Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
⊢ ( 𝑎 = 𝑐 → ( 𝑎 +no 𝑏 ) = ( 𝑐 +no 𝑏 ) ) |
2 |
1
|
eleq1d |
⊢ ( 𝑎 = 𝑐 → ( ( 𝑎 +no 𝑏 ) ∈ On ↔ ( 𝑐 +no 𝑏 ) ∈ On ) ) |
3 |
|
sneq |
⊢ ( 𝑎 = 𝑐 → { 𝑎 } = { 𝑐 } ) |
4 |
3
|
xpeq1d |
⊢ ( 𝑎 = 𝑐 → ( { 𝑎 } × 𝑏 ) = ( { 𝑐 } × 𝑏 ) ) |
5 |
4
|
imaeq2d |
⊢ ( 𝑎 = 𝑐 → ( +no “ ( { 𝑎 } × 𝑏 ) ) = ( +no “ ( { 𝑐 } × 𝑏 ) ) ) |
6 |
5
|
sseq1d |
⊢ ( 𝑎 = 𝑐 → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ) ) |
7 |
|
xpeq1 |
⊢ ( 𝑎 = 𝑐 → ( 𝑎 × { 𝑏 } ) = ( 𝑐 × { 𝑏 } ) ) |
8 |
7
|
imaeq2d |
⊢ ( 𝑎 = 𝑐 → ( +no “ ( 𝑎 × { 𝑏 } ) ) = ( +no “ ( 𝑐 × { 𝑏 } ) ) ) |
9 |
8
|
sseq1d |
⊢ ( 𝑎 = 𝑐 → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) ) |
10 |
6 9
|
anbi12d |
⊢ ( 𝑎 = 𝑐 → ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) ) ) |
11 |
10
|
rabbidv |
⊢ ( 𝑎 = 𝑐 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
12 |
11
|
inteqd |
⊢ ( 𝑎 = 𝑐 → ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
13 |
1 12
|
eqeq12d |
⊢ ( 𝑎 = 𝑐 → ( ( 𝑎 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ↔ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) |
14 |
2 13
|
anbi12d |
⊢ ( 𝑎 = 𝑐 → ( ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑏 = 𝑑 → ( 𝑐 +no 𝑏 ) = ( 𝑐 +no 𝑑 ) ) |
16 |
15
|
eleq1d |
⊢ ( 𝑏 = 𝑑 → ( ( 𝑐 +no 𝑏 ) ∈ On ↔ ( 𝑐 +no 𝑑 ) ∈ On ) ) |
17 |
|
xpeq2 |
⊢ ( 𝑏 = 𝑑 → ( { 𝑐 } × 𝑏 ) = ( { 𝑐 } × 𝑑 ) ) |
18 |
17
|
imaeq2d |
⊢ ( 𝑏 = 𝑑 → ( +no “ ( { 𝑐 } × 𝑏 ) ) = ( +no “ ( { 𝑐 } × 𝑑 ) ) ) |
19 |
18
|
sseq1d |
⊢ ( 𝑏 = 𝑑 → ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ) ) |
20 |
|
sneq |
⊢ ( 𝑏 = 𝑑 → { 𝑏 } = { 𝑑 } ) |
21 |
20
|
xpeq2d |
⊢ ( 𝑏 = 𝑑 → ( 𝑐 × { 𝑏 } ) = ( 𝑐 × { 𝑑 } ) ) |
22 |
21
|
imaeq2d |
⊢ ( 𝑏 = 𝑑 → ( +no “ ( 𝑐 × { 𝑏 } ) ) = ( +no “ ( 𝑐 × { 𝑑 } ) ) ) |
23 |
22
|
sseq1d |
⊢ ( 𝑏 = 𝑑 → ( ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) ) |
24 |
19 23
|
anbi12d |
⊢ ( 𝑏 = 𝑑 → ( ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) ) ) |
25 |
24
|
rabbidv |
⊢ ( 𝑏 = 𝑑 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) |
26 |
25
|
inteqd |
⊢ ( 𝑏 = 𝑑 → ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) |
27 |
15 26
|
eqeq12d |
⊢ ( 𝑏 = 𝑑 → ( ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ↔ ( 𝑐 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) |
28 |
16 27
|
anbi12d |
⊢ ( 𝑏 = 𝑑 → ( ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) ) |
29 |
|
oveq1 |
⊢ ( 𝑎 = 𝑐 → ( 𝑎 +no 𝑑 ) = ( 𝑐 +no 𝑑 ) ) |
30 |
29
|
eleq1d |
⊢ ( 𝑎 = 𝑐 → ( ( 𝑎 +no 𝑑 ) ∈ On ↔ ( 𝑐 +no 𝑑 ) ∈ On ) ) |
31 |
3
|
xpeq1d |
⊢ ( 𝑎 = 𝑐 → ( { 𝑎 } × 𝑑 ) = ( { 𝑐 } × 𝑑 ) ) |
32 |
31
|
imaeq2d |
⊢ ( 𝑎 = 𝑐 → ( +no “ ( { 𝑎 } × 𝑑 ) ) = ( +no “ ( { 𝑐 } × 𝑑 ) ) ) |
33 |
32
|
sseq1d |
⊢ ( 𝑎 = 𝑐 → ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ) ) |
34 |
|
xpeq1 |
⊢ ( 𝑎 = 𝑐 → ( 𝑎 × { 𝑑 } ) = ( 𝑐 × { 𝑑 } ) ) |
35 |
34
|
imaeq2d |
⊢ ( 𝑎 = 𝑐 → ( +no “ ( 𝑎 × { 𝑑 } ) ) = ( +no “ ( 𝑐 × { 𝑑 } ) ) ) |
36 |
35
|
sseq1d |
⊢ ( 𝑎 = 𝑐 → ( ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) ) |
37 |
33 36
|
anbi12d |
⊢ ( 𝑎 = 𝑐 → ( ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) ) ) |
38 |
37
|
rabbidv |
⊢ ( 𝑎 = 𝑐 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) |
39 |
38
|
inteqd |
⊢ ( 𝑎 = 𝑐 → ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) |
40 |
29 39
|
eqeq12d |
⊢ ( 𝑎 = 𝑐 → ( ( 𝑎 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ↔ ( 𝑐 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) |
41 |
30 40
|
anbi12d |
⊢ ( 𝑎 = 𝑐 → ( ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) ) |
42 |
|
oveq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 +no 𝑏 ) = ( 𝐴 +no 𝑏 ) ) |
43 |
42
|
eleq1d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 +no 𝑏 ) ∈ On ↔ ( 𝐴 +no 𝑏 ) ∈ On ) ) |
44 |
|
sneq |
⊢ ( 𝑎 = 𝐴 → { 𝑎 } = { 𝐴 } ) |
45 |
44
|
xpeq1d |
⊢ ( 𝑎 = 𝐴 → ( { 𝑎 } × 𝑏 ) = ( { 𝐴 } × 𝑏 ) ) |
46 |
45
|
imaeq2d |
⊢ ( 𝑎 = 𝐴 → ( +no “ ( { 𝑎 } × 𝑏 ) ) = ( +no “ ( { 𝐴 } × 𝑏 ) ) ) |
47 |
46
|
sseq1d |
⊢ ( 𝑎 = 𝐴 → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ) ) |
48 |
|
xpeq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 × { 𝑏 } ) = ( 𝐴 × { 𝑏 } ) ) |
49 |
48
|
imaeq2d |
⊢ ( 𝑎 = 𝐴 → ( +no “ ( 𝑎 × { 𝑏 } ) ) = ( +no “ ( 𝐴 × { 𝑏 } ) ) ) |
50 |
49
|
sseq1d |
⊢ ( 𝑎 = 𝐴 → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) ) |
51 |
47 50
|
anbi12d |
⊢ ( 𝑎 = 𝐴 → ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) ) ) |
52 |
51
|
rabbidv |
⊢ ( 𝑎 = 𝐴 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
53 |
52
|
inteqd |
⊢ ( 𝑎 = 𝐴 → ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
54 |
42 53
|
eqeq12d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ↔ ( 𝐴 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) |
55 |
43 54
|
anbi12d |
⊢ ( 𝑎 = 𝐴 → ( ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝐴 +no 𝑏 ) ∈ On ∧ ( 𝐴 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) ) |
56 |
|
oveq2 |
⊢ ( 𝑏 = 𝐵 → ( 𝐴 +no 𝑏 ) = ( 𝐴 +no 𝐵 ) ) |
57 |
56
|
eleq1d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 +no 𝑏 ) ∈ On ↔ ( 𝐴 +no 𝐵 ) ∈ On ) ) |
58 |
|
xpeq2 |
⊢ ( 𝑏 = 𝐵 → ( { 𝐴 } × 𝑏 ) = ( { 𝐴 } × 𝐵 ) ) |
59 |
58
|
imaeq2d |
⊢ ( 𝑏 = 𝐵 → ( +no “ ( { 𝐴 } × 𝑏 ) ) = ( +no “ ( { 𝐴 } × 𝐵 ) ) ) |
60 |
59
|
sseq1d |
⊢ ( 𝑏 = 𝐵 → ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ) ) |
61 |
|
sneq |
⊢ ( 𝑏 = 𝐵 → { 𝑏 } = { 𝐵 } ) |
62 |
61
|
xpeq2d |
⊢ ( 𝑏 = 𝐵 → ( 𝐴 × { 𝑏 } ) = ( 𝐴 × { 𝐵 } ) ) |
63 |
62
|
imaeq2d |
⊢ ( 𝑏 = 𝐵 → ( +no “ ( 𝐴 × { 𝑏 } ) ) = ( +no “ ( 𝐴 × { 𝐵 } ) ) ) |
64 |
63
|
sseq1d |
⊢ ( 𝑏 = 𝐵 → ( ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) ) |
65 |
60 64
|
anbi12d |
⊢ ( 𝑏 = 𝐵 → ( ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) ) ) |
66 |
65
|
rabbidv |
⊢ ( 𝑏 = 𝐵 → { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) |
67 |
66
|
inteqd |
⊢ ( 𝑏 = 𝐵 → ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) |
68 |
56 67
|
eqeq12d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ↔ ( 𝐴 +no 𝐵 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) ) |
69 |
57 68
|
anbi12d |
⊢ ( 𝑏 = 𝐵 → ( ( ( 𝐴 +no 𝑏 ) ∈ On ∧ ( 𝐴 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ↔ ( ( 𝐴 +no 𝐵 ) ∈ On ∧ ( 𝐴 +no 𝐵 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) ) ) |
70 |
|
simpl |
⊢ ( ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) → ( 𝑐 +no 𝑏 ) ∈ On ) |
71 |
70
|
ralimi |
⊢ ( ∀ 𝑐 ∈ 𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) → ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ) |
72 |
71
|
3ad2ant2 |
⊢ ( ( ∀ 𝑐 ∈ 𝑎 ∀ 𝑑 ∈ 𝑏 ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑐 ∈ 𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑑 ∈ 𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) → ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ) |
73 |
|
simpl |
⊢ ( ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) → ( 𝑎 +no 𝑑 ) ∈ On ) |
74 |
73
|
ralimi |
⊢ ( ∀ 𝑑 ∈ 𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) → ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) |
75 |
74
|
3ad2ant3 |
⊢ ( ( ∀ 𝑐 ∈ 𝑎 ∀ 𝑑 ∈ 𝑏 ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑐 ∈ 𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑑 ∈ 𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) → ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) |
76 |
72 75
|
jca |
⊢ ( ( ∀ 𝑐 ∈ 𝑎 ∀ 𝑑 ∈ 𝑏 ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑐 ∈ 𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑑 ∈ 𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) → ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) |
77 |
|
df-nadd |
⊢ +no = frecs ( { 〈 𝑝 , 𝑞 〉 ∣ ( 𝑝 ∈ ( On × On ) ∧ 𝑞 ∈ ( On × On ) ∧ ( ( ( 1st ‘ 𝑝 ) E ( 1st ‘ 𝑞 ) ∨ ( 1st ‘ 𝑝 ) = ( 1st ‘ 𝑞 ) ) ∧ ( ( 2nd ‘ 𝑝 ) E ( 2nd ‘ 𝑞 ) ∨ ( 2nd ‘ 𝑝 ) = ( 2nd ‘ 𝑞 ) ) ∧ 𝑝 ≠ 𝑞 ) ) } , ( On × On ) , ( 𝑡 ∈ V , 𝑓 ∈ V ↦ ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } ) ) |
78 |
77
|
on2recsov |
⊢ ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( 𝑎 +no 𝑏 ) = ( 〈 𝑎 , 𝑏 〉 ( 𝑡 ∈ V , 𝑓 ∈ V ↦ ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } ) ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) ) ) |
79 |
78
|
adantr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 +no 𝑏 ) = ( 〈 𝑎 , 𝑏 〉 ( 𝑡 ∈ V , 𝑓 ∈ V ↦ ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } ) ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) ) ) |
80 |
|
opex |
⊢ 〈 𝑎 , 𝑏 〉 ∈ V |
81 |
|
naddfn |
⊢ +no Fn ( On × On ) |
82 |
|
fnfun |
⊢ ( +no Fn ( On × On ) → Fun +no ) |
83 |
81 82
|
ax-mp |
⊢ Fun +no |
84 |
|
vex |
⊢ 𝑎 ∈ V |
85 |
84
|
sucex |
⊢ suc 𝑎 ∈ V |
86 |
|
vex |
⊢ 𝑏 ∈ V |
87 |
86
|
sucex |
⊢ suc 𝑏 ∈ V |
88 |
85 87
|
xpex |
⊢ ( suc 𝑎 × suc 𝑏 ) ∈ V |
89 |
88
|
difexi |
⊢ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ∈ V |
90 |
|
resfunexg |
⊢ ( ( Fun +no ∧ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ∈ V ) → ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) ∈ V ) |
91 |
83 89 90
|
mp2an |
⊢ ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) ∈ V |
92 |
|
eloni |
⊢ ( 𝑏 ∈ On → Ord 𝑏 ) |
93 |
92
|
ad2antlr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → Ord 𝑏 ) |
94 |
|
ordirr |
⊢ ( Ord 𝑏 → ¬ 𝑏 ∈ 𝑏 ) |
95 |
93 94
|
syl |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ¬ 𝑏 ∈ 𝑏 ) |
96 |
95
|
olcd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ¬ 𝑎 ∈ { 𝑎 } ∨ ¬ 𝑏 ∈ 𝑏 ) ) |
97 |
|
ianor |
⊢ ( ¬ ( 𝑎 ∈ { 𝑎 } ∧ 𝑏 ∈ 𝑏 ) ↔ ( ¬ 𝑎 ∈ { 𝑎 } ∨ ¬ 𝑏 ∈ 𝑏 ) ) |
98 |
|
opelxp |
⊢ ( 〈 𝑎 , 𝑏 〉 ∈ ( { 𝑎 } × 𝑏 ) ↔ ( 𝑎 ∈ { 𝑎 } ∧ 𝑏 ∈ 𝑏 ) ) |
99 |
97 98
|
xchnxbir |
⊢ ( ¬ 〈 𝑎 , 𝑏 〉 ∈ ( { 𝑎 } × 𝑏 ) ↔ ( ¬ 𝑎 ∈ { 𝑎 } ∨ ¬ 𝑏 ∈ 𝑏 ) ) |
100 |
96 99
|
sylibr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ¬ 〈 𝑎 , 𝑏 〉 ∈ ( { 𝑎 } × 𝑏 ) ) |
101 |
84
|
sucid |
⊢ 𝑎 ∈ suc 𝑎 |
102 |
|
snssi |
⊢ ( 𝑎 ∈ suc 𝑎 → { 𝑎 } ⊆ suc 𝑎 ) |
103 |
101 102
|
ax-mp |
⊢ { 𝑎 } ⊆ suc 𝑎 |
104 |
|
sssucid |
⊢ 𝑏 ⊆ suc 𝑏 |
105 |
|
xpss12 |
⊢ ( ( { 𝑎 } ⊆ suc 𝑎 ∧ 𝑏 ⊆ suc 𝑏 ) → ( { 𝑎 } × 𝑏 ) ⊆ ( suc 𝑎 × suc 𝑏 ) ) |
106 |
103 104 105
|
mp2an |
⊢ ( { 𝑎 } × 𝑏 ) ⊆ ( suc 𝑎 × suc 𝑏 ) |
107 |
|
ssdifsn |
⊢ ( ( { 𝑎 } × 𝑏 ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ↔ ( ( { 𝑎 } × 𝑏 ) ⊆ ( suc 𝑎 × suc 𝑏 ) ∧ ¬ 〈 𝑎 , 𝑏 〉 ∈ ( { 𝑎 } × 𝑏 ) ) ) |
108 |
106 107
|
mpbiran |
⊢ ( ( { 𝑎 } × 𝑏 ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ↔ ¬ 〈 𝑎 , 𝑏 〉 ∈ ( { 𝑎 } × 𝑏 ) ) |
109 |
100 108
|
sylibr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( { 𝑎 } × 𝑏 ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) |
110 |
|
resima2 |
⊢ ( ( { 𝑎 } × 𝑏 ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) → ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) = ( +no “ ( { 𝑎 } × 𝑏 ) ) ) |
111 |
109 110
|
syl |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) = ( +no “ ( { 𝑎 } × 𝑏 ) ) ) |
112 |
111
|
sseq1d |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ) ) |
113 |
|
eloni |
⊢ ( 𝑎 ∈ On → Ord 𝑎 ) |
114 |
113
|
ad2antrr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → Ord 𝑎 ) |
115 |
|
ordirr |
⊢ ( Ord 𝑎 → ¬ 𝑎 ∈ 𝑎 ) |
116 |
114 115
|
syl |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ¬ 𝑎 ∈ 𝑎 ) |
117 |
116
|
orcd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ¬ 𝑎 ∈ 𝑎 ∨ ¬ 𝑏 ∈ { 𝑏 } ) ) |
118 |
|
ianor |
⊢ ( ¬ ( 𝑎 ∈ 𝑎 ∧ 𝑏 ∈ { 𝑏 } ) ↔ ( ¬ 𝑎 ∈ 𝑎 ∨ ¬ 𝑏 ∈ { 𝑏 } ) ) |
119 |
|
opelxp |
⊢ ( 〈 𝑎 , 𝑏 〉 ∈ ( 𝑎 × { 𝑏 } ) ↔ ( 𝑎 ∈ 𝑎 ∧ 𝑏 ∈ { 𝑏 } ) ) |
120 |
118 119
|
xchnxbir |
⊢ ( ¬ 〈 𝑎 , 𝑏 〉 ∈ ( 𝑎 × { 𝑏 } ) ↔ ( ¬ 𝑎 ∈ 𝑎 ∨ ¬ 𝑏 ∈ { 𝑏 } ) ) |
121 |
117 120
|
sylibr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ¬ 〈 𝑎 , 𝑏 〉 ∈ ( 𝑎 × { 𝑏 } ) ) |
122 |
|
sssucid |
⊢ 𝑎 ⊆ suc 𝑎 |
123 |
86
|
sucid |
⊢ 𝑏 ∈ suc 𝑏 |
124 |
|
snssi |
⊢ ( 𝑏 ∈ suc 𝑏 → { 𝑏 } ⊆ suc 𝑏 ) |
125 |
123 124
|
ax-mp |
⊢ { 𝑏 } ⊆ suc 𝑏 |
126 |
|
xpss12 |
⊢ ( ( 𝑎 ⊆ suc 𝑎 ∧ { 𝑏 } ⊆ suc 𝑏 ) → ( 𝑎 × { 𝑏 } ) ⊆ ( suc 𝑎 × suc 𝑏 ) ) |
127 |
122 125 126
|
mp2an |
⊢ ( 𝑎 × { 𝑏 } ) ⊆ ( suc 𝑎 × suc 𝑏 ) |
128 |
|
ssdifsn |
⊢ ( ( 𝑎 × { 𝑏 } ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ↔ ( ( 𝑎 × { 𝑏 } ) ⊆ ( suc 𝑎 × suc 𝑏 ) ∧ ¬ 〈 𝑎 , 𝑏 〉 ∈ ( 𝑎 × { 𝑏 } ) ) ) |
129 |
127 128
|
mpbiran |
⊢ ( ( 𝑎 × { 𝑏 } ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ↔ ¬ 〈 𝑎 , 𝑏 〉 ∈ ( 𝑎 × { 𝑏 } ) ) |
130 |
121 129
|
sylibr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 × { 𝑏 } ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) |
131 |
|
resima2 |
⊢ ( ( 𝑎 × { 𝑏 } ) ⊆ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) → ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) = ( +no “ ( 𝑎 × { 𝑏 } ) ) ) |
132 |
130 131
|
syl |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) = ( +no “ ( 𝑎 × { 𝑏 } ) ) ) |
133 |
132
|
sseq1d |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) |
134 |
112 133
|
anbi12d |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) ) |
135 |
134
|
rabbidv |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
136 |
135
|
inteqd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∩ { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
137 |
|
simprr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) |
138 |
|
oveq1 |
⊢ ( 𝑡 = 𝑎 → ( 𝑡 +no 𝑑 ) = ( 𝑎 +no 𝑑 ) ) |
139 |
138
|
eleq1d |
⊢ ( 𝑡 = 𝑎 → ( ( 𝑡 +no 𝑑 ) ∈ On ↔ ( 𝑎 +no 𝑑 ) ∈ On ) ) |
140 |
139
|
ralbidv |
⊢ ( 𝑡 = 𝑎 → ( ∀ 𝑑 ∈ 𝑏 ( 𝑡 +no 𝑑 ) ∈ On ↔ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) |
141 |
84 140
|
ralsn |
⊢ ( ∀ 𝑡 ∈ { 𝑎 } ∀ 𝑑 ∈ 𝑏 ( 𝑡 +no 𝑑 ) ∈ On ↔ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) |
142 |
137 141
|
sylibr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∀ 𝑡 ∈ { 𝑎 } ∀ 𝑑 ∈ 𝑏 ( 𝑡 +no 𝑑 ) ∈ On ) |
143 |
|
snssi |
⊢ ( 𝑎 ∈ On → { 𝑎 } ⊆ On ) |
144 |
|
onss |
⊢ ( 𝑏 ∈ On → 𝑏 ⊆ On ) |
145 |
|
xpss12 |
⊢ ( ( { 𝑎 } ⊆ On ∧ 𝑏 ⊆ On ) → ( { 𝑎 } × 𝑏 ) ⊆ ( On × On ) ) |
146 |
143 144 145
|
syl2an |
⊢ ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( { 𝑎 } × 𝑏 ) ⊆ ( On × On ) ) |
147 |
146
|
adantr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( { 𝑎 } × 𝑏 ) ⊆ ( On × On ) ) |
148 |
81
|
fndmi |
⊢ dom +no = ( On × On ) |
149 |
147 148
|
sseqtrrdi |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( { 𝑎 } × 𝑏 ) ⊆ dom +no ) |
150 |
|
funimassov |
⊢ ( ( Fun +no ∧ ( { 𝑎 } × 𝑏 ) ⊆ dom +no ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ On ↔ ∀ 𝑡 ∈ { 𝑎 } ∀ 𝑑 ∈ 𝑏 ( 𝑡 +no 𝑑 ) ∈ On ) ) |
151 |
83 149 150
|
sylancr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ On ↔ ∀ 𝑡 ∈ { 𝑎 } ∀ 𝑑 ∈ 𝑏 ( 𝑡 +no 𝑑 ) ∈ On ) ) |
152 |
142 151
|
mpbird |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ On ) |
153 |
|
simprl |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ) |
154 |
|
oveq2 |
⊢ ( 𝑡 = 𝑏 → ( 𝑐 +no 𝑡 ) = ( 𝑐 +no 𝑏 ) ) |
155 |
154
|
eleq1d |
⊢ ( 𝑡 = 𝑏 → ( ( 𝑐 +no 𝑡 ) ∈ On ↔ ( 𝑐 +no 𝑏 ) ∈ On ) ) |
156 |
86 155
|
ralsn |
⊢ ( ∀ 𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ↔ ( 𝑐 +no 𝑏 ) ∈ On ) |
157 |
156
|
ralbii |
⊢ ( ∀ 𝑐 ∈ 𝑎 ∀ 𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ↔ ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ) |
158 |
153 157
|
sylibr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∀ 𝑐 ∈ 𝑎 ∀ 𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ) |
159 |
|
onss |
⊢ ( 𝑎 ∈ On → 𝑎 ⊆ On ) |
160 |
|
snssi |
⊢ ( 𝑏 ∈ On → { 𝑏 } ⊆ On ) |
161 |
|
xpss12 |
⊢ ( ( 𝑎 ⊆ On ∧ { 𝑏 } ⊆ On ) → ( 𝑎 × { 𝑏 } ) ⊆ ( On × On ) ) |
162 |
159 160 161
|
syl2an |
⊢ ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( 𝑎 × { 𝑏 } ) ⊆ ( On × On ) ) |
163 |
162
|
adantr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 × { 𝑏 } ) ⊆ ( On × On ) ) |
164 |
163 148
|
sseqtrrdi |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 × { 𝑏 } ) ⊆ dom +no ) |
165 |
|
funimassov |
⊢ ( ( Fun +no ∧ ( 𝑎 × { 𝑏 } ) ⊆ dom +no ) → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ On ↔ ∀ 𝑐 ∈ 𝑎 ∀ 𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ) ) |
166 |
83 164 165
|
sylancr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ On ↔ ∀ 𝑐 ∈ 𝑎 ∀ 𝑡 ∈ { 𝑏 } ( 𝑐 +no 𝑡 ) ∈ On ) ) |
167 |
158 166
|
mpbird |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ On ) |
168 |
152 167
|
unssd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ On ) |
169 |
|
ssorduni |
⊢ ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ On → Ord ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) |
170 |
168 169
|
syl |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → Ord ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) |
171 |
|
snex |
⊢ { 𝑎 } ∈ V |
172 |
171 86
|
xpex |
⊢ ( { 𝑎 } × 𝑏 ) ∈ V |
173 |
|
funimaexg |
⊢ ( ( Fun +no ∧ ( { 𝑎 } × 𝑏 ) ∈ V ) → ( +no “ ( { 𝑎 } × 𝑏 ) ) ∈ V ) |
174 |
83 172 173
|
mp2an |
⊢ ( +no “ ( { 𝑎 } × 𝑏 ) ) ∈ V |
175 |
|
snex |
⊢ { 𝑏 } ∈ V |
176 |
84 175
|
xpex |
⊢ ( 𝑎 × { 𝑏 } ) ∈ V |
177 |
|
funimaexg |
⊢ ( ( Fun +no ∧ ( 𝑎 × { 𝑏 } ) ∈ V ) → ( +no “ ( 𝑎 × { 𝑏 } ) ) ∈ V ) |
178 |
83 176 177
|
mp2an |
⊢ ( +no “ ( 𝑎 × { 𝑏 } ) ) ∈ V |
179 |
174 178
|
unex |
⊢ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ V |
180 |
179
|
uniex |
⊢ ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ V |
181 |
180
|
elon |
⊢ ( ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ↔ Ord ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) |
182 |
170 181
|
sylibr |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ) |
183 |
|
sucelon |
⊢ ( ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ↔ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ) |
184 |
182 183
|
sylib |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ) |
185 |
|
onsucuni |
⊢ ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ On → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) |
186 |
168 185
|
syl |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) |
187 |
186
|
unssad |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) |
188 |
186
|
unssbd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) |
189 |
|
sseq2 |
⊢ ( 𝑥 = suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) → ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) ) |
190 |
|
sseq2 |
⊢ ( 𝑥 = suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) → ( ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) ) |
191 |
189 190
|
anbi12d |
⊢ ( 𝑥 = suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) → ( ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) ) ) |
192 |
191
|
rspcev |
⊢ ( ( suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∈ On ∧ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ suc ∪ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ∪ ( +no “ ( 𝑎 × { 𝑏 } ) ) ) ) ) → ∃ 𝑥 ∈ On ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) |
193 |
184 187 188 192
|
syl12anc |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∃ 𝑥 ∈ On ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) |
194 |
|
onintrab2 |
⊢ ( ∃ 𝑥 ∈ On ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ∈ On ) |
195 |
193 194
|
sylib |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ∈ On ) |
196 |
136 195
|
eqeltrd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ∩ { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ∈ On ) |
197 |
84 86
|
op1std |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( 1st ‘ 𝑡 ) = 𝑎 ) |
198 |
197
|
sneqd |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → { ( 1st ‘ 𝑡 ) } = { 𝑎 } ) |
199 |
84 86
|
op2ndd |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( 2nd ‘ 𝑡 ) = 𝑏 ) |
200 |
198 199
|
xpeq12d |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) = ( { 𝑎 } × 𝑏 ) ) |
201 |
200
|
imaeq2d |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) = ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ) |
202 |
201
|
sseq1d |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ↔ ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ) ) |
203 |
199
|
sneqd |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → { ( 2nd ‘ 𝑡 ) } = { 𝑏 } ) |
204 |
197 203
|
xpeq12d |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) = ( 𝑎 × { 𝑏 } ) ) |
205 |
204
|
imaeq2d |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) = ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ) |
206 |
205
|
sseq1d |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ↔ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) |
207 |
202 206
|
anbi12d |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ( ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) ↔ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) ) |
208 |
207
|
rabbidv |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
209 |
208
|
inteqd |
⊢ ( 𝑡 = 〈 𝑎 , 𝑏 〉 → ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
210 |
|
imaeq1 |
⊢ ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) → ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) = ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ) |
211 |
210
|
sseq1d |
⊢ ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) → ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ↔ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ) ) |
212 |
|
imaeq1 |
⊢ ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) → ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) = ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ) |
213 |
212
|
sseq1d |
⊢ ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) → ( ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ↔ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) |
214 |
211 213
|
anbi12d |
⊢ ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) → ( ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ↔ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) ) ) |
215 |
214
|
rabbidv |
⊢ ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) → { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
216 |
215
|
inteqd |
⊢ ( 𝑓 = ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) → ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
217 |
|
eqid |
⊢ ( 𝑡 ∈ V , 𝑓 ∈ V ↦ ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } ) = ( 𝑡 ∈ V , 𝑓 ∈ V ↦ ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } ) |
218 |
209 216 217
|
ovmpog |
⊢ ( ( 〈 𝑎 , 𝑏 〉 ∈ V ∧ ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) ∈ V ∧ ∩ { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ∈ On ) → ( 〈 𝑎 , 𝑏 〉 ( 𝑡 ∈ V , 𝑓 ∈ V ↦ ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } ) ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) ) = ∩ { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
219 |
80 91 196 218
|
mp3an12i |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 〈 𝑎 , 𝑏 〉 ( 𝑡 ∈ V , 𝑓 ∈ V ↦ ∩ { 𝑥 ∈ On ∣ ( ( 𝑓 “ ( { ( 1st ‘ 𝑡 ) } × ( 2nd ‘ 𝑡 ) ) ) ⊆ 𝑥 ∧ ( 𝑓 “ ( ( 1st ‘ 𝑡 ) × { ( 2nd ‘ 𝑡 ) } ) ) ⊆ 𝑥 ) } ) ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) ) = ∩ { 𝑥 ∈ On ∣ ( ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( ( +no ↾ ( ( suc 𝑎 × suc 𝑏 ) ∖ { 〈 𝑎 , 𝑏 〉 } ) ) “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
220 |
79 219 136
|
3eqtrd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) |
221 |
220 195
|
eqeltrd |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( 𝑎 +no 𝑏 ) ∈ On ) |
222 |
221 220
|
jca |
⊢ ( ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) ∧ ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) ) → ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) |
223 |
222
|
ex |
⊢ ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( ( ∀ 𝑐 ∈ 𝑎 ( 𝑐 +no 𝑏 ) ∈ On ∧ ∀ 𝑑 ∈ 𝑏 ( 𝑎 +no 𝑑 ) ∈ On ) → ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) ) |
224 |
76 223
|
syl5 |
⊢ ( ( 𝑎 ∈ On ∧ 𝑏 ∈ On ) → ( ( ∀ 𝑐 ∈ 𝑎 ∀ 𝑑 ∈ 𝑏 ( ( 𝑐 +no 𝑑 ) ∈ On ∧ ( 𝑐 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑐 ∈ 𝑎 ( ( 𝑐 +no 𝑏 ) ∈ On ∧ ( 𝑐 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑐 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑐 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ∧ ∀ 𝑑 ∈ 𝑏 ( ( 𝑎 +no 𝑑 ) ∈ On ∧ ( 𝑎 +no 𝑑 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑑 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑑 } ) ) ⊆ 𝑥 ) } ) ) → ( ( 𝑎 +no 𝑏 ) ∈ On ∧ ( 𝑎 +no 𝑏 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝑎 } × 𝑏 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝑎 × { 𝑏 } ) ) ⊆ 𝑥 ) } ) ) ) |
225 |
14 28 41 55 69 224
|
on2ind |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +no 𝐵 ) ∈ On ∧ ( 𝐴 +no 𝐵 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) ) |