Step |
Hyp |
Ref |
Expression |
1 |
|
soseq.1 |
⊢ 𝑅 Or ( 𝐴 ∪ { ∅ } ) |
2 |
|
soseq.2 |
⊢ 𝐹 = { 𝑓 ∣ ∃ 𝑥 ∈ On 𝑓 : 𝑥 ⟶ 𝐴 } |
3 |
|
soseq.3 |
⊢ 𝑆 = { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) } |
4 |
|
soseq.4 |
⊢ ¬ ∅ ∈ 𝐴 |
5 |
|
sopo |
⊢ ( 𝑅 Or ( 𝐴 ∪ { ∅ } ) → 𝑅 Po ( 𝐴 ∪ { ∅ } ) ) |
6 |
1 5
|
ax-mp |
⊢ 𝑅 Po ( 𝐴 ∪ { ∅ } ) |
7 |
6 2 3
|
poseq |
⊢ 𝑆 Po 𝐹 |
8 |
|
eleq1w |
⊢ ( 𝑓 = 𝑎 → ( 𝑓 ∈ 𝐹 ↔ 𝑎 ∈ 𝐹 ) ) |
9 |
8
|
anbi1d |
⊢ ( 𝑓 = 𝑎 → ( ( 𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ↔ ( 𝑎 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ) ) |
10 |
|
fveq1 |
⊢ ( 𝑓 = 𝑎 → ( 𝑓 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ) |
11 |
10
|
eqeq1d |
⊢ ( 𝑓 = 𝑎 → ( ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ↔ ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ) ) |
12 |
11
|
ralbidv |
⊢ ( 𝑓 = 𝑎 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ) ) |
13 |
|
fveq1 |
⊢ ( 𝑓 = 𝑎 → ( 𝑓 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑥 ) ) |
14 |
13
|
breq1d |
⊢ ( 𝑓 = 𝑎 → ( ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ↔ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) |
15 |
12 14
|
anbi12d |
⊢ ( 𝑓 = 𝑎 → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ) |
16 |
15
|
rexbidv |
⊢ ( 𝑓 = 𝑎 → ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ) |
17 |
9 16
|
anbi12d |
⊢ ( 𝑓 = 𝑎 → ( ( ( 𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ↔ ( ( 𝑎 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ) ) |
18 |
|
eleq1w |
⊢ ( 𝑔 = 𝑏 → ( 𝑔 ∈ 𝐹 ↔ 𝑏 ∈ 𝐹 ) ) |
19 |
18
|
anbi2d |
⊢ ( 𝑔 = 𝑏 → ( ( 𝑎 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ↔ ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) ) ) |
20 |
|
fveq1 |
⊢ ( 𝑔 = 𝑏 → ( 𝑔 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) |
21 |
20
|
eqeq2d |
⊢ ( 𝑔 = 𝑏 → ( ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ↔ ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) ) |
22 |
21
|
ralbidv |
⊢ ( 𝑔 = 𝑏 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) ) |
23 |
|
fveq1 |
⊢ ( 𝑔 = 𝑏 → ( 𝑔 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) |
24 |
23
|
breq2d |
⊢ ( 𝑔 = 𝑏 → ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ↔ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ) |
25 |
22 24
|
anbi12d |
⊢ ( 𝑔 = 𝑏 → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ) ) |
26 |
25
|
rexbidv |
⊢ ( 𝑔 = 𝑏 → ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ) ) |
27 |
19 26
|
anbi12d |
⊢ ( 𝑔 = 𝑏 → ( ( ( 𝑎 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ↔ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ) ) ) |
28 |
17 27 3
|
brabg |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( 𝑎 𝑆 𝑏 ↔ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ) ) ) |
29 |
28
|
bianabs |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( 𝑎 𝑆 𝑏 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ) ) |
30 |
|
eleq1w |
⊢ ( 𝑓 = 𝑏 → ( 𝑓 ∈ 𝐹 ↔ 𝑏 ∈ 𝐹 ) ) |
31 |
30
|
anbi1d |
⊢ ( 𝑓 = 𝑏 → ( ( 𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ↔ ( 𝑏 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ) ) |
32 |
|
fveq1 |
⊢ ( 𝑓 = 𝑏 → ( 𝑓 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) |
33 |
32
|
eqeq1d |
⊢ ( 𝑓 = 𝑏 → ( ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ↔ ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ) ) |
34 |
33
|
ralbidv |
⊢ ( 𝑓 = 𝑏 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ) ) |
35 |
|
fveq1 |
⊢ ( 𝑓 = 𝑏 → ( 𝑓 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) |
36 |
35
|
breq1d |
⊢ ( 𝑓 = 𝑏 → ( ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ↔ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) |
37 |
34 36
|
anbi12d |
⊢ ( 𝑓 = 𝑏 → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ) |
38 |
37
|
rexbidv |
⊢ ( 𝑓 = 𝑏 → ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ) |
39 |
31 38
|
anbi12d |
⊢ ( 𝑓 = 𝑏 → ( ( ( 𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ↔ ( ( 𝑏 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ) ) |
40 |
|
eleq1w |
⊢ ( 𝑔 = 𝑎 → ( 𝑔 ∈ 𝐹 ↔ 𝑎 ∈ 𝐹 ) ) |
41 |
40
|
anbi2d |
⊢ ( 𝑔 = 𝑎 → ( ( 𝑏 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ↔ ( 𝑏 ∈ 𝐹 ∧ 𝑎 ∈ 𝐹 ) ) ) |
42 |
|
fveq1 |
⊢ ( 𝑔 = 𝑎 → ( 𝑔 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ) |
43 |
42
|
eqeq2d |
⊢ ( 𝑔 = 𝑎 → ( ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ↔ ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ) ) |
44 |
43
|
ralbidv |
⊢ ( 𝑔 = 𝑎 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ) ) |
45 |
|
fveq1 |
⊢ ( 𝑔 = 𝑎 → ( 𝑔 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑥 ) ) |
46 |
45
|
breq2d |
⊢ ( 𝑔 = 𝑎 → ( ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ↔ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) |
47 |
44 46
|
anbi12d |
⊢ ( 𝑔 = 𝑎 → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
48 |
47
|
rexbidv |
⊢ ( 𝑔 = 𝑎 → ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
49 |
41 48
|
anbi12d |
⊢ ( 𝑔 = 𝑎 → ( ( ( 𝑏 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) ↔ ( ( 𝑏 ∈ 𝐹 ∧ 𝑎 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) ) |
50 |
39 49 3
|
brabg |
⊢ ( ( 𝑏 ∈ 𝐹 ∧ 𝑎 ∈ 𝐹 ) → ( 𝑏 𝑆 𝑎 ↔ ( ( 𝑏 ∈ 𝐹 ∧ 𝑎 ∈ 𝐹 ) ∧ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) ) |
51 |
50
|
bianabs |
⊢ ( ( 𝑏 ∈ 𝐹 ∧ 𝑎 ∈ 𝐹 ) → ( 𝑏 𝑆 𝑎 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
52 |
51
|
ancoms |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( 𝑏 𝑆 𝑎 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
53 |
29 52
|
orbi12d |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ( 𝑎 𝑆 𝑏 ∨ 𝑏 𝑆 𝑎 ) ↔ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) ) |
54 |
53
|
notbid |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ¬ ( 𝑎 𝑆 𝑏 ∨ 𝑏 𝑆 𝑎 ) ↔ ¬ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) ) |
55 |
|
ralinexa |
⊢ ( ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ¬ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ↔ ¬ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
56 |
|
andi |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ↔ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
57 |
|
eqcom |
⊢ ( ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ↔ ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ) |
58 |
57
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ) |
59 |
58
|
anbi1i |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) |
60 |
59
|
orbi2i |
⊢ ( ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ↔ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
61 |
56 60
|
bitri |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ↔ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
62 |
61
|
rexbii |
⊢ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ↔ ∃ 𝑥 ∈ On ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
63 |
|
r19.43 |
⊢ ( ∃ 𝑥 ∈ On ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ↔ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
64 |
62 63
|
bitri |
⊢ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ↔ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
65 |
55 64
|
xchbinx |
⊢ ( ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ¬ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ↔ ¬ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
66 |
|
feq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑓 : 𝑥 ⟶ 𝐴 ↔ 𝑓 : 𝑦 ⟶ 𝐴 ) ) |
67 |
66
|
cbvrexvw |
⊢ ( ∃ 𝑥 ∈ On 𝑓 : 𝑥 ⟶ 𝐴 ↔ ∃ 𝑦 ∈ On 𝑓 : 𝑦 ⟶ 𝐴 ) |
68 |
67
|
abbii |
⊢ { 𝑓 ∣ ∃ 𝑥 ∈ On 𝑓 : 𝑥 ⟶ 𝐴 } = { 𝑓 ∣ ∃ 𝑦 ∈ On 𝑓 : 𝑦 ⟶ 𝐴 } |
69 |
2 68
|
eqtri |
⊢ 𝐹 = { 𝑓 ∣ ∃ 𝑦 ∈ On 𝑓 : 𝑦 ⟶ 𝐴 } |
70 |
69
|
orderseqlem |
⊢ ( 𝑎 ∈ 𝐹 → ( 𝑎 ‘ 𝑥 ) ∈ ( 𝐴 ∪ { ∅ } ) ) |
71 |
69
|
orderseqlem |
⊢ ( 𝑏 ∈ 𝐹 → ( 𝑏 ‘ 𝑥 ) ∈ ( 𝐴 ∪ { ∅ } ) ) |
72 |
|
sotrieq |
⊢ ( ( 𝑅 Or ( 𝐴 ∪ { ∅ } ) ∧ ( ( 𝑎 ‘ 𝑥 ) ∈ ( 𝐴 ∪ { ∅ } ) ∧ ( 𝑏 ‘ 𝑥 ) ∈ ( 𝐴 ∪ { ∅ } ) ) ) → ( ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ↔ ¬ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
73 |
1 72
|
mpan |
⊢ ( ( ( 𝑎 ‘ 𝑥 ) ∈ ( 𝐴 ∪ { ∅ } ) ∧ ( 𝑏 ‘ 𝑥 ) ∈ ( 𝐴 ∪ { ∅ } ) ) → ( ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ↔ ¬ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
74 |
70 71 73
|
syl2an |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ↔ ¬ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) |
75 |
74
|
imbi2d |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ¬ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) ) |
76 |
75
|
ralbidv |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ¬ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) ) ) |
77 |
|
vex |
⊢ 𝑦 ∈ V |
78 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑎 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑦 ) ) |
79 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑏 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑦 ) ) |
80 |
78 79
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ↔ ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) ) |
81 |
77 80
|
sbcie |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ↔ ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) |
82 |
81
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝑥 [ 𝑦 / 𝑥 ] ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) |
83 |
82
|
imbi1i |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 [ 𝑦 / 𝑥 ] ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) |
84 |
83
|
ralbii |
⊢ ( ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 [ 𝑦 / 𝑥 ] ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) |
85 |
|
tfisg |
⊢ ( ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 [ 𝑦 / 𝑥 ] ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) → ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) |
86 |
84 85
|
sylbir |
⊢ ( ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) → ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) |
87 |
|
vex |
⊢ 𝑎 ∈ V |
88 |
|
feq1 |
⊢ ( 𝑓 = 𝑎 → ( 𝑓 : 𝑥 ⟶ 𝐴 ↔ 𝑎 : 𝑥 ⟶ 𝐴 ) ) |
89 |
88
|
rexbidv |
⊢ ( 𝑓 = 𝑎 → ( ∃ 𝑥 ∈ On 𝑓 : 𝑥 ⟶ 𝐴 ↔ ∃ 𝑥 ∈ On 𝑎 : 𝑥 ⟶ 𝐴 ) ) |
90 |
87 89 2
|
elab2 |
⊢ ( 𝑎 ∈ 𝐹 ↔ ∃ 𝑥 ∈ On 𝑎 : 𝑥 ⟶ 𝐴 ) |
91 |
|
feq2 |
⊢ ( 𝑥 = 𝑝 → ( 𝑎 : 𝑥 ⟶ 𝐴 ↔ 𝑎 : 𝑝 ⟶ 𝐴 ) ) |
92 |
91
|
cbvrexvw |
⊢ ( ∃ 𝑥 ∈ On 𝑎 : 𝑥 ⟶ 𝐴 ↔ ∃ 𝑝 ∈ On 𝑎 : 𝑝 ⟶ 𝐴 ) |
93 |
90 92
|
bitri |
⊢ ( 𝑎 ∈ 𝐹 ↔ ∃ 𝑝 ∈ On 𝑎 : 𝑝 ⟶ 𝐴 ) |
94 |
|
vex |
⊢ 𝑏 ∈ V |
95 |
|
feq1 |
⊢ ( 𝑓 = 𝑏 → ( 𝑓 : 𝑥 ⟶ 𝐴 ↔ 𝑏 : 𝑥 ⟶ 𝐴 ) ) |
96 |
95
|
rexbidv |
⊢ ( 𝑓 = 𝑏 → ( ∃ 𝑥 ∈ On 𝑓 : 𝑥 ⟶ 𝐴 ↔ ∃ 𝑥 ∈ On 𝑏 : 𝑥 ⟶ 𝐴 ) ) |
97 |
94 96 2
|
elab2 |
⊢ ( 𝑏 ∈ 𝐹 ↔ ∃ 𝑥 ∈ On 𝑏 : 𝑥 ⟶ 𝐴 ) |
98 |
|
feq2 |
⊢ ( 𝑥 = 𝑞 → ( 𝑏 : 𝑥 ⟶ 𝐴 ↔ 𝑏 : 𝑞 ⟶ 𝐴 ) ) |
99 |
98
|
cbvrexvw |
⊢ ( ∃ 𝑥 ∈ On 𝑏 : 𝑥 ⟶ 𝐴 ↔ ∃ 𝑞 ∈ On 𝑏 : 𝑞 ⟶ 𝐴 ) |
100 |
97 99
|
bitri |
⊢ ( 𝑏 ∈ 𝐹 ↔ ∃ 𝑞 ∈ On 𝑏 : 𝑞 ⟶ 𝐴 ) |
101 |
93 100
|
anbi12i |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) ↔ ( ∃ 𝑝 ∈ On 𝑎 : 𝑝 ⟶ 𝐴 ∧ ∃ 𝑞 ∈ On 𝑏 : 𝑞 ⟶ 𝐴 ) ) |
102 |
|
reeanv |
⊢ ( ∃ 𝑝 ∈ On ∃ 𝑞 ∈ On ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ↔ ( ∃ 𝑝 ∈ On 𝑎 : 𝑝 ⟶ 𝐴 ∧ ∃ 𝑞 ∈ On 𝑏 : 𝑞 ⟶ 𝐴 ) ) |
103 |
101 102
|
bitr4i |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) ↔ ∃ 𝑝 ∈ On ∃ 𝑞 ∈ On ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) |
104 |
|
onss |
⊢ ( 𝑞 ∈ On → 𝑞 ⊆ On ) |
105 |
|
ssralv |
⊢ ( 𝑞 ⊆ On → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) |
106 |
104 105
|
syl |
⊢ ( 𝑞 ∈ On → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) |
107 |
106
|
ad2antlr |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) |
108 |
|
fveq2 |
⊢ ( 𝑥 = 𝑝 → ( 𝑎 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑝 ) ) |
109 |
|
fveq2 |
⊢ ( 𝑥 = 𝑝 → ( 𝑏 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑝 ) ) |
110 |
108 109
|
eqeq12d |
⊢ ( 𝑥 = 𝑝 → ( ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ↔ ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) ) ) |
111 |
110
|
rspcv |
⊢ ( 𝑝 ∈ 𝑞 → ( ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) ) ) |
112 |
111
|
a1i |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( 𝑝 ∈ 𝑞 → ( ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) ) ) ) |
113 |
|
ffvelrn |
⊢ ( ( 𝑏 : 𝑞 ⟶ 𝐴 ∧ 𝑝 ∈ 𝑞 ) → ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 ) |
114 |
|
fdm |
⊢ ( 𝑎 : 𝑝 ⟶ 𝐴 → dom 𝑎 = 𝑝 ) |
115 |
|
eloni |
⊢ ( 𝑝 ∈ On → Ord 𝑝 ) |
116 |
|
ordirr |
⊢ ( Ord 𝑝 → ¬ 𝑝 ∈ 𝑝 ) |
117 |
115 116
|
syl |
⊢ ( 𝑝 ∈ On → ¬ 𝑝 ∈ 𝑝 ) |
118 |
|
eleq2 |
⊢ ( dom 𝑎 = 𝑝 → ( 𝑝 ∈ dom 𝑎 ↔ 𝑝 ∈ 𝑝 ) ) |
119 |
118
|
notbid |
⊢ ( dom 𝑎 = 𝑝 → ( ¬ 𝑝 ∈ dom 𝑎 ↔ ¬ 𝑝 ∈ 𝑝 ) ) |
120 |
119
|
biimparc |
⊢ ( ( ¬ 𝑝 ∈ 𝑝 ∧ dom 𝑎 = 𝑝 ) → ¬ 𝑝 ∈ dom 𝑎 ) |
121 |
117 120
|
sylan |
⊢ ( ( 𝑝 ∈ On ∧ dom 𝑎 = 𝑝 ) → ¬ 𝑝 ∈ dom 𝑎 ) |
122 |
|
ndmfv |
⊢ ( ¬ 𝑝 ∈ dom 𝑎 → ( 𝑎 ‘ 𝑝 ) = ∅ ) |
123 |
|
eqtr2 |
⊢ ( ( ( 𝑎 ‘ 𝑝 ) = ∅ ∧ ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) ) → ∅ = ( 𝑏 ‘ 𝑝 ) ) |
124 |
|
eleq1 |
⊢ ( ∅ = ( 𝑏 ‘ 𝑝 ) → ( ∅ ∈ 𝐴 ↔ ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 ) ) |
125 |
124
|
biimprd |
⊢ ( ∅ = ( 𝑏 ‘ 𝑝 ) → ( ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 → ∅ ∈ 𝐴 ) ) |
126 |
123 125
|
syl |
⊢ ( ( ( 𝑎 ‘ 𝑝 ) = ∅ ∧ ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) ) → ( ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 → ∅ ∈ 𝐴 ) ) |
127 |
126
|
ex |
⊢ ( ( 𝑎 ‘ 𝑝 ) = ∅ → ( ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) → ( ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 → ∅ ∈ 𝐴 ) ) ) |
128 |
121 122 127
|
3syl |
⊢ ( ( 𝑝 ∈ On ∧ dom 𝑎 = 𝑝 ) → ( ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) → ( ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 → ∅ ∈ 𝐴 ) ) ) |
129 |
128
|
com23 |
⊢ ( ( 𝑝 ∈ On ∧ dom 𝑎 = 𝑝 ) → ( ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 → ( ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) → ∅ ∈ 𝐴 ) ) ) |
130 |
114 129
|
sylan2 |
⊢ ( ( 𝑝 ∈ On ∧ 𝑎 : 𝑝 ⟶ 𝐴 ) → ( ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 → ( ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) → ∅ ∈ 𝐴 ) ) ) |
131 |
130
|
adantlr |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ 𝑎 : 𝑝 ⟶ 𝐴 ) → ( ( 𝑏 ‘ 𝑝 ) ∈ 𝐴 → ( ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) → ∅ ∈ 𝐴 ) ) ) |
132 |
113 131
|
syl5 |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ 𝑎 : 𝑝 ⟶ 𝐴 ) → ( ( 𝑏 : 𝑞 ⟶ 𝐴 ∧ 𝑝 ∈ 𝑞 ) → ( ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) → ∅ ∈ 𝐴 ) ) ) |
133 |
132
|
exp4b |
⊢ ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) → ( 𝑎 : 𝑝 ⟶ 𝐴 → ( 𝑏 : 𝑞 ⟶ 𝐴 → ( 𝑝 ∈ 𝑞 → ( ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) → ∅ ∈ 𝐴 ) ) ) ) ) |
134 |
133
|
imp32 |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( 𝑝 ∈ 𝑞 → ( ( 𝑎 ‘ 𝑝 ) = ( 𝑏 ‘ 𝑝 ) → ∅ ∈ 𝐴 ) ) ) |
135 |
112 134
|
syldd |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( 𝑝 ∈ 𝑞 → ( ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ∅ ∈ 𝐴 ) ) ) |
136 |
135
|
com23 |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑝 ∈ 𝑞 → ∅ ∈ 𝐴 ) ) ) |
137 |
136
|
imp |
⊢ ( ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) ∧ ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) → ( 𝑝 ∈ 𝑞 → ∅ ∈ 𝐴 ) ) |
138 |
4 137
|
mtoi |
⊢ ( ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) ∧ ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) → ¬ 𝑝 ∈ 𝑞 ) |
139 |
138
|
ex |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ 𝑞 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ¬ 𝑝 ∈ 𝑞 ) ) |
140 |
107 139
|
syld |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ¬ 𝑝 ∈ 𝑞 ) ) |
141 |
|
onss |
⊢ ( 𝑝 ∈ On → 𝑝 ⊆ On ) |
142 |
|
ssralv |
⊢ ( 𝑝 ⊆ On → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) |
143 |
141 142
|
syl |
⊢ ( 𝑝 ∈ On → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) |
144 |
143
|
ad2antrr |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) |
145 |
|
fveq2 |
⊢ ( 𝑥 = 𝑞 → ( 𝑎 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑞 ) ) |
146 |
|
fveq2 |
⊢ ( 𝑥 = 𝑞 → ( 𝑏 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑞 ) ) |
147 |
145 146
|
eqeq12d |
⊢ ( 𝑥 = 𝑞 → ( ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ↔ ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) ) ) |
148 |
147
|
rspcv |
⊢ ( 𝑞 ∈ 𝑝 → ( ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) ) ) |
149 |
148
|
a1i |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( 𝑞 ∈ 𝑝 → ( ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) ) ) ) |
150 |
|
ffvelrn |
⊢ ( ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑞 ∈ 𝑝 ) → ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 ) |
151 |
|
fdm |
⊢ ( 𝑏 : 𝑞 ⟶ 𝐴 → dom 𝑏 = 𝑞 ) |
152 |
|
eloni |
⊢ ( 𝑞 ∈ On → Ord 𝑞 ) |
153 |
|
ordirr |
⊢ ( Ord 𝑞 → ¬ 𝑞 ∈ 𝑞 ) |
154 |
152 153
|
syl |
⊢ ( 𝑞 ∈ On → ¬ 𝑞 ∈ 𝑞 ) |
155 |
|
eleq2 |
⊢ ( dom 𝑏 = 𝑞 → ( 𝑞 ∈ dom 𝑏 ↔ 𝑞 ∈ 𝑞 ) ) |
156 |
155
|
notbid |
⊢ ( dom 𝑏 = 𝑞 → ( ¬ 𝑞 ∈ dom 𝑏 ↔ ¬ 𝑞 ∈ 𝑞 ) ) |
157 |
156
|
biimparc |
⊢ ( ( ¬ 𝑞 ∈ 𝑞 ∧ dom 𝑏 = 𝑞 ) → ¬ 𝑞 ∈ dom 𝑏 ) |
158 |
|
ndmfv |
⊢ ( ¬ 𝑞 ∈ dom 𝑏 → ( 𝑏 ‘ 𝑞 ) = ∅ ) |
159 |
157 158
|
syl |
⊢ ( ( ¬ 𝑞 ∈ 𝑞 ∧ dom 𝑏 = 𝑞 ) → ( 𝑏 ‘ 𝑞 ) = ∅ ) |
160 |
154 159
|
sylan |
⊢ ( ( 𝑞 ∈ On ∧ dom 𝑏 = 𝑞 ) → ( 𝑏 ‘ 𝑞 ) = ∅ ) |
161 |
|
eqtr |
⊢ ( ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) ∧ ( 𝑏 ‘ 𝑞 ) = ∅ ) → ( 𝑎 ‘ 𝑞 ) = ∅ ) |
162 |
|
eleq1 |
⊢ ( ( 𝑎 ‘ 𝑞 ) = ∅ → ( ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 ↔ ∅ ∈ 𝐴 ) ) |
163 |
162
|
biimpd |
⊢ ( ( 𝑎 ‘ 𝑞 ) = ∅ → ( ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 → ∅ ∈ 𝐴 ) ) |
164 |
161 163
|
syl |
⊢ ( ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) ∧ ( 𝑏 ‘ 𝑞 ) = ∅ ) → ( ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 → ∅ ∈ 𝐴 ) ) |
165 |
164
|
expcom |
⊢ ( ( 𝑏 ‘ 𝑞 ) = ∅ → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ( ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 → ∅ ∈ 𝐴 ) ) ) |
166 |
165
|
com23 |
⊢ ( ( 𝑏 ‘ 𝑞 ) = ∅ → ( ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ∅ ∈ 𝐴 ) ) ) |
167 |
160 166
|
syl |
⊢ ( ( 𝑞 ∈ On ∧ dom 𝑏 = 𝑞 ) → ( ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ∅ ∈ 𝐴 ) ) ) |
168 |
167
|
adantll |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ dom 𝑏 = 𝑞 ) → ( ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ∅ ∈ 𝐴 ) ) ) |
169 |
151 168
|
sylan2 |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) → ( ( 𝑎 ‘ 𝑞 ) ∈ 𝐴 → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ∅ ∈ 𝐴 ) ) ) |
170 |
150 169
|
syl5 |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) → ( ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑞 ∈ 𝑝 ) → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ∅ ∈ 𝐴 ) ) ) |
171 |
170
|
exp4b |
⊢ ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) → ( 𝑏 : 𝑞 ⟶ 𝐴 → ( 𝑎 : 𝑝 ⟶ 𝐴 → ( 𝑞 ∈ 𝑝 → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ∅ ∈ 𝐴 ) ) ) ) ) |
172 |
171
|
com23 |
⊢ ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) → ( 𝑎 : 𝑝 ⟶ 𝐴 → ( 𝑏 : 𝑞 ⟶ 𝐴 → ( 𝑞 ∈ 𝑝 → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ∅ ∈ 𝐴 ) ) ) ) ) |
173 |
172
|
imp32 |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( 𝑞 ∈ 𝑝 → ( ( 𝑎 ‘ 𝑞 ) = ( 𝑏 ‘ 𝑞 ) → ∅ ∈ 𝐴 ) ) ) |
174 |
149 173
|
syldd |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( 𝑞 ∈ 𝑝 → ( ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ∅ ∈ 𝐴 ) ) ) |
175 |
174
|
com23 |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑞 ∈ 𝑝 → ∅ ∈ 𝐴 ) ) ) |
176 |
175
|
imp |
⊢ ( ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) ∧ ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) → ( 𝑞 ∈ 𝑝 → ∅ ∈ 𝐴 ) ) |
177 |
4 176
|
mtoi |
⊢ ( ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) ∧ ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) → ¬ 𝑞 ∈ 𝑝 ) |
178 |
177
|
ex |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ¬ 𝑞 ∈ 𝑝 ) ) |
179 |
144 178
|
syld |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ¬ 𝑞 ∈ 𝑝 ) ) |
180 |
140 179
|
jcad |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( ¬ 𝑝 ∈ 𝑞 ∧ ¬ 𝑞 ∈ 𝑝 ) ) ) |
181 |
|
ordtri3or |
⊢ ( ( Ord 𝑝 ∧ Ord 𝑞 ) → ( 𝑝 ∈ 𝑞 ∨ 𝑝 = 𝑞 ∨ 𝑞 ∈ 𝑝 ) ) |
182 |
115 152 181
|
syl2an |
⊢ ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) → ( 𝑝 ∈ 𝑞 ∨ 𝑝 = 𝑞 ∨ 𝑞 ∈ 𝑝 ) ) |
183 |
182
|
adantr |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( 𝑝 ∈ 𝑞 ∨ 𝑝 = 𝑞 ∨ 𝑞 ∈ 𝑝 ) ) |
184 |
|
3orel13 |
⊢ ( ( ¬ 𝑝 ∈ 𝑞 ∧ ¬ 𝑞 ∈ 𝑝 ) → ( ( 𝑝 ∈ 𝑞 ∨ 𝑝 = 𝑞 ∨ 𝑞 ∈ 𝑝 ) → 𝑝 = 𝑞 ) ) |
185 |
180 183 184
|
syl6ci |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → 𝑝 = 𝑞 ) ) |
186 |
185 144
|
jcad |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → ( 𝑝 = 𝑞 ∧ ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) ) |
187 |
|
ffn |
⊢ ( 𝑎 : 𝑝 ⟶ 𝐴 → 𝑎 Fn 𝑝 ) |
188 |
|
ffn |
⊢ ( 𝑏 : 𝑞 ⟶ 𝐴 → 𝑏 Fn 𝑞 ) |
189 |
|
eqfnfv2 |
⊢ ( ( 𝑎 Fn 𝑝 ∧ 𝑏 Fn 𝑞 ) → ( 𝑎 = 𝑏 ↔ ( 𝑝 = 𝑞 ∧ ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) ) |
190 |
187 188 189
|
syl2an |
⊢ ( ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) → ( 𝑎 = 𝑏 ↔ ( 𝑝 = 𝑞 ∧ ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) ) |
191 |
190
|
adantl |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( 𝑎 = 𝑏 ↔ ( 𝑝 = 𝑞 ∧ ∀ 𝑥 ∈ 𝑝 ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) ) ) |
192 |
186 191
|
sylibrd |
⊢ ( ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) ∧ ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → 𝑎 = 𝑏 ) ) |
193 |
192
|
ex |
⊢ ( ( 𝑝 ∈ On ∧ 𝑞 ∈ On ) → ( ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → 𝑎 = 𝑏 ) ) ) |
194 |
193
|
rexlimivv |
⊢ ( ∃ 𝑝 ∈ On ∃ 𝑞 ∈ On ( 𝑎 : 𝑝 ⟶ 𝐴 ∧ 𝑏 : 𝑞 ⟶ 𝐴 ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → 𝑎 = 𝑏 ) ) |
195 |
103 194
|
sylbi |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ∀ 𝑥 ∈ On ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) → 𝑎 = 𝑏 ) ) |
196 |
86 195
|
syl5 |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑥 ) ) → 𝑎 = 𝑏 ) ) |
197 |
76 196
|
sylbird |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ∀ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) → ¬ ( ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ∨ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) → 𝑎 = 𝑏 ) ) |
198 |
65 197
|
syl5bir |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ¬ ( ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑎 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ∧ ( 𝑎 ‘ 𝑥 ) 𝑅 ( 𝑏 ‘ 𝑥 ) ) ∨ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝑏 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ∧ ( 𝑏 ‘ 𝑥 ) 𝑅 ( 𝑎 ‘ 𝑥 ) ) ) → 𝑎 = 𝑏 ) ) |
199 |
54 198
|
sylbid |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ¬ ( 𝑎 𝑆 𝑏 ∨ 𝑏 𝑆 𝑎 ) → 𝑎 = 𝑏 ) ) |
200 |
199
|
orrd |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( ( 𝑎 𝑆 𝑏 ∨ 𝑏 𝑆 𝑎 ) ∨ 𝑎 = 𝑏 ) ) |
201 |
|
3orcomb |
⊢ ( ( 𝑎 𝑆 𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏 𝑆 𝑎 ) ↔ ( 𝑎 𝑆 𝑏 ∨ 𝑏 𝑆 𝑎 ∨ 𝑎 = 𝑏 ) ) |
202 |
|
df-3or |
⊢ ( ( 𝑎 𝑆 𝑏 ∨ 𝑏 𝑆 𝑎 ∨ 𝑎 = 𝑏 ) ↔ ( ( 𝑎 𝑆 𝑏 ∨ 𝑏 𝑆 𝑎 ) ∨ 𝑎 = 𝑏 ) ) |
203 |
201 202
|
bitr2i |
⊢ ( ( ( 𝑎 𝑆 𝑏 ∨ 𝑏 𝑆 𝑎 ) ∨ 𝑎 = 𝑏 ) ↔ ( 𝑎 𝑆 𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏 𝑆 𝑎 ) ) |
204 |
200 203
|
sylib |
⊢ ( ( 𝑎 ∈ 𝐹 ∧ 𝑏 ∈ 𝐹 ) → ( 𝑎 𝑆 𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏 𝑆 𝑎 ) ) |
205 |
204
|
rgen2 |
⊢ ∀ 𝑎 ∈ 𝐹 ∀ 𝑏 ∈ 𝐹 ( 𝑎 𝑆 𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏 𝑆 𝑎 ) |
206 |
|
df-so |
⊢ ( 𝑆 Or 𝐹 ↔ ( 𝑆 Po 𝐹 ∧ ∀ 𝑎 ∈ 𝐹 ∀ 𝑏 ∈ 𝐹 ( 𝑎 𝑆 𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏 𝑆 𝑎 ) ) ) |
207 |
7 205 206
|
mpbir2an |
⊢ 𝑆 Or 𝐹 |