Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑥 = ∅ → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( ( 𝐴 +o 𝐵 ) +o ∅ ) ) |
2 |
|
oveq2 |
⊢ ( 𝑥 = ∅ → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o ∅ ) ) |
3 |
2
|
oveq2d |
⊢ ( 𝑥 = ∅ → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ( 𝐴 +o ( 𝐵 +o ∅ ) ) ) |
4 |
1 3
|
eqeq12d |
⊢ ( 𝑥 = ∅ → ( ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) ↔ ( ( 𝐴 +o 𝐵 ) +o ∅ ) = ( 𝐴 +o ( 𝐵 +o ∅ ) ) ) ) |
5 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) ) |
6 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o 𝑦 ) ) |
7 |
6
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
8 |
5 7
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) ↔ ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( ( 𝐴 +o 𝐵 ) +o suc 𝑦 ) ) |
10 |
|
oveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o suc 𝑦 ) ) |
11 |
10
|
oveq2d |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) ) |
12 |
9 11
|
eqeq12d |
⊢ ( 𝑥 = suc 𝑦 → ( ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) ↔ ( ( 𝐴 +o 𝐵 ) +o suc 𝑦 ) = ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( ( 𝐴 +o 𝐵 ) +o 𝐶 ) ) |
14 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o 𝐶 ) ) |
15 |
14
|
oveq2d |
⊢ ( 𝑥 = 𝐶 → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ( 𝐴 +o ( 𝐵 +o 𝐶 ) ) ) |
16 |
13 15
|
eqeq12d |
⊢ ( 𝑥 = 𝐶 → ( ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) ↔ ( ( 𝐴 +o 𝐵 ) +o 𝐶 ) = ( 𝐴 +o ( 𝐵 +o 𝐶 ) ) ) ) |
17 |
|
oacl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) ∈ On ) |
18 |
|
oa0 |
⊢ ( ( 𝐴 +o 𝐵 ) ∈ On → ( ( 𝐴 +o 𝐵 ) +o ∅ ) = ( 𝐴 +o 𝐵 ) ) |
19 |
17 18
|
syl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) +o ∅ ) = ( 𝐴 +o 𝐵 ) ) |
20 |
|
oa0 |
⊢ ( 𝐵 ∈ On → ( 𝐵 +o ∅ ) = 𝐵 ) |
21 |
20
|
oveq2d |
⊢ ( 𝐵 ∈ On → ( 𝐴 +o ( 𝐵 +o ∅ ) ) = ( 𝐴 +o 𝐵 ) ) |
22 |
21
|
adantl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o ( 𝐵 +o ∅ ) ) = ( 𝐴 +o 𝐵 ) ) |
23 |
19 22
|
eqtr4d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) +o ∅ ) = ( 𝐴 +o ( 𝐵 +o ∅ ) ) ) |
24 |
|
suceq |
⊢ ( ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → suc ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = suc ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
25 |
|
oasuc |
⊢ ( ( ( 𝐴 +o 𝐵 ) ∈ On ∧ 𝑦 ∈ On ) → ( ( 𝐴 +o 𝐵 ) +o suc 𝑦 ) = suc ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) ) |
26 |
17 25
|
sylan |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑦 ∈ On ) → ( ( 𝐴 +o 𝐵 ) +o suc 𝑦 ) = suc ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) ) |
27 |
|
oasuc |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐵 +o suc 𝑦 ) = suc ( 𝐵 +o 𝑦 ) ) |
28 |
27
|
oveq2d |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) = ( 𝐴 +o suc ( 𝐵 +o 𝑦 ) ) ) |
29 |
28
|
adantl |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) ) → ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) = ( 𝐴 +o suc ( 𝐵 +o 𝑦 ) ) ) |
30 |
|
oacl |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐵 +o 𝑦 ) ∈ On ) |
31 |
|
oasuc |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 +o 𝑦 ) ∈ On ) → ( 𝐴 +o suc ( 𝐵 +o 𝑦 ) ) = suc ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
32 |
30 31
|
sylan2 |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) ) → ( 𝐴 +o suc ( 𝐵 +o 𝑦 ) ) = suc ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
33 |
29 32
|
eqtrd |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) ) → ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) = suc ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
34 |
33
|
anassrs |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑦 ∈ On ) → ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) = suc ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
35 |
26 34
|
eqeq12d |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑦 ∈ On ) → ( ( ( 𝐴 +o 𝐵 ) +o suc 𝑦 ) = ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) ↔ suc ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = suc ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) |
36 |
24 35
|
syl5ibr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑦 ∈ On ) → ( ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → ( ( 𝐴 +o 𝐵 ) +o suc 𝑦 ) = ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) ) ) |
37 |
36
|
expcom |
⊢ ( 𝑦 ∈ On → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → ( ( 𝐴 +o 𝐵 ) +o suc 𝑦 ) = ( 𝐴 +o ( 𝐵 +o suc 𝑦 ) ) ) ) ) |
38 |
|
iuneq2 |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → ∪ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
39 |
38
|
adantl |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) → ∪ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
40 |
|
vex |
⊢ 𝑥 ∈ V |
41 |
|
oalim |
⊢ ( ( ( 𝐴 +o 𝐵 ) ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) ) |
42 |
40 41
|
mpanr1 |
⊢ ( ( ( 𝐴 +o 𝐵 ) ∈ On ∧ Lim 𝑥 ) → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) ) |
43 |
17 42
|
sylan |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ Lim 𝑥 ) → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) ) |
44 |
43
|
ancoms |
⊢ ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) ) |
45 |
44
|
adantr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) ) |
46 |
|
oalimcl |
⊢ ( ( 𝐵 ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) → Lim ( 𝐵 +o 𝑥 ) ) |
47 |
40 46
|
mpanr1 |
⊢ ( ( 𝐵 ∈ On ∧ Lim 𝑥 ) → Lim ( 𝐵 +o 𝑥 ) ) |
48 |
47
|
ancoms |
⊢ ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) → Lim ( 𝐵 +o 𝑥 ) ) |
49 |
|
ovex |
⊢ ( 𝐵 +o 𝑥 ) ∈ V |
50 |
|
oalim |
⊢ ( ( 𝐴 ∈ On ∧ ( ( 𝐵 +o 𝑥 ) ∈ V ∧ Lim ( 𝐵 +o 𝑥 ) ) ) → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ) |
51 |
49 50
|
mpanr1 |
⊢ ( ( 𝐴 ∈ On ∧ Lim ( 𝐵 +o 𝑥 ) ) → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ) |
52 |
48 51
|
sylan2 |
⊢ ( ( 𝐴 ∈ On ∧ ( Lim 𝑥 ∧ 𝐵 ∈ On ) ) → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ) |
53 |
|
limelon |
⊢ ( ( 𝑥 ∈ V ∧ Lim 𝑥 ) → 𝑥 ∈ On ) |
54 |
40 53
|
mpan |
⊢ ( Lim 𝑥 → 𝑥 ∈ On ) |
55 |
|
oacl |
⊢ ( ( 𝐵 ∈ On ∧ 𝑥 ∈ On ) → ( 𝐵 +o 𝑥 ) ∈ On ) |
56 |
55
|
ancoms |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 +o 𝑥 ) ∈ On ) |
57 |
|
onelon |
⊢ ( ( ( 𝐵 +o 𝑥 ) ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → 𝑧 ∈ On ) |
58 |
57
|
ex |
⊢ ( ( 𝐵 +o 𝑥 ) ∈ On → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → 𝑧 ∈ On ) ) |
59 |
56 58
|
syl |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → 𝑧 ∈ On ) ) |
60 |
59
|
adantld |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → 𝑧 ∈ On ) ) |
61 |
60
|
adantl |
⊢ ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → 𝑧 ∈ On ) ) |
62 |
|
0ellim |
⊢ ( Lim 𝑥 → ∅ ∈ 𝑥 ) |
63 |
|
onelss |
⊢ ( 𝐵 ∈ On → ( 𝑧 ∈ 𝐵 → 𝑧 ⊆ 𝐵 ) ) |
64 |
20
|
sseq2d |
⊢ ( 𝐵 ∈ On → ( 𝑧 ⊆ ( 𝐵 +o ∅ ) ↔ 𝑧 ⊆ 𝐵 ) ) |
65 |
63 64
|
sylibrd |
⊢ ( 𝐵 ∈ On → ( 𝑧 ∈ 𝐵 → 𝑧 ⊆ ( 𝐵 +o ∅ ) ) ) |
66 |
65
|
imp |
⊢ ( ( 𝐵 ∈ On ∧ 𝑧 ∈ 𝐵 ) → 𝑧 ⊆ ( 𝐵 +o ∅ ) ) |
67 |
|
oveq2 |
⊢ ( 𝑦 = ∅ → ( 𝐵 +o 𝑦 ) = ( 𝐵 +o ∅ ) ) |
68 |
67
|
sseq2d |
⊢ ( 𝑦 = ∅ → ( 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ↔ 𝑧 ⊆ ( 𝐵 +o ∅ ) ) ) |
69 |
68
|
rspcev |
⊢ ( ( ∅ ∈ 𝑥 ∧ 𝑧 ⊆ ( 𝐵 +o ∅ ) ) → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) |
70 |
62 66 69
|
syl2an |
⊢ ( ( Lim 𝑥 ∧ ( 𝐵 ∈ On ∧ 𝑧 ∈ 𝐵 ) ) → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) |
71 |
70
|
expr |
⊢ ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) → ( 𝑧 ∈ 𝐵 → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
72 |
71
|
adantrl |
⊢ ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑧 ∈ 𝐵 → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
73 |
72
|
adantrr |
⊢ ( ( Lim 𝑥 ∧ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑧 ∈ On ) ) ) → ( 𝑧 ∈ 𝐵 → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
74 |
|
oawordex |
⊢ ( ( 𝐵 ∈ On ∧ 𝑧 ∈ On ) → ( 𝐵 ⊆ 𝑧 ↔ ∃ 𝑦 ∈ On ( 𝐵 +o 𝑦 ) = 𝑧 ) ) |
75 |
74
|
ad2ant2l |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑧 ∈ On ) ) → ( 𝐵 ⊆ 𝑧 ↔ ∃ 𝑦 ∈ On ( 𝐵 +o 𝑦 ) = 𝑧 ) ) |
76 |
|
oaord |
⊢ ( ( 𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑦 ∈ 𝑥 ↔ ( 𝐵 +o 𝑦 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
77 |
76
|
3expb |
⊢ ( ( 𝑦 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑦 ∈ 𝑥 ↔ ( 𝐵 +o 𝑦 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
78 |
|
eleq1 |
⊢ ( ( 𝐵 +o 𝑦 ) = 𝑧 → ( ( 𝐵 +o 𝑦 ) ∈ ( 𝐵 +o 𝑥 ) ↔ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ) |
79 |
77 78
|
sylan9bb |
⊢ ( ( ( 𝑦 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( 𝐵 +o 𝑦 ) = 𝑧 ) → ( 𝑦 ∈ 𝑥 ↔ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ) |
80 |
79
|
an32s |
⊢ ( ( ( 𝑦 ∈ On ∧ ( 𝐵 +o 𝑦 ) = 𝑧 ) ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑦 ∈ 𝑥 ↔ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ) |
81 |
80
|
biimpar |
⊢ ( ( ( ( 𝑦 ∈ On ∧ ( 𝐵 +o 𝑦 ) = 𝑧 ) ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → 𝑦 ∈ 𝑥 ) |
82 |
|
eqimss2 |
⊢ ( ( 𝐵 +o 𝑦 ) = 𝑧 → 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) |
83 |
82
|
ad3antlr |
⊢ ( ( ( ( 𝑦 ∈ On ∧ ( 𝐵 +o 𝑦 ) = 𝑧 ) ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) |
84 |
81 83
|
jca |
⊢ ( ( ( ( 𝑦 ∈ On ∧ ( 𝐵 +o 𝑦 ) = 𝑧 ) ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( 𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
85 |
84
|
anasss |
⊢ ( ( ( 𝑦 ∈ On ∧ ( 𝐵 +o 𝑦 ) = 𝑧 ) ∧ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ) → ( 𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
86 |
85
|
expcom |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( ( 𝑦 ∈ On ∧ ( 𝐵 +o 𝑦 ) = 𝑧 ) → ( 𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) ) |
87 |
86
|
reximdv2 |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( ∃ 𝑦 ∈ On ( 𝐵 +o 𝑦 ) = 𝑧 → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
88 |
87
|
adantrr |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑧 ∈ On ) ) → ( ∃ 𝑦 ∈ On ( 𝐵 +o 𝑦 ) = 𝑧 → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
89 |
75 88
|
sylbid |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑧 ∈ On ) ) → ( 𝐵 ⊆ 𝑧 → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
90 |
89
|
adantl |
⊢ ( ( Lim 𝑥 ∧ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑧 ∈ On ) ) ) → ( 𝐵 ⊆ 𝑧 → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) |
91 |
|
eloni |
⊢ ( 𝑧 ∈ On → Ord 𝑧 ) |
92 |
|
eloni |
⊢ ( 𝐵 ∈ On → Ord 𝐵 ) |
93 |
|
ordtri2or |
⊢ ( ( Ord 𝑧 ∧ Ord 𝐵 ) → ( 𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧 ) ) |
94 |
91 92 93
|
syl2anr |
⊢ ( ( 𝐵 ∈ On ∧ 𝑧 ∈ On ) → ( 𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧 ) ) |
95 |
94
|
ad2ant2l |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑧 ∈ On ) ) → ( 𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧 ) ) |
96 |
95
|
adantl |
⊢ ( ( Lim 𝑥 ∧ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑧 ∈ On ) ) ) → ( 𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧 ) ) |
97 |
73 90 96
|
mpjaod |
⊢ ( ( Lim 𝑥 ∧ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑧 ∈ On ) ) ) → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) |
98 |
97
|
exp45 |
⊢ ( Lim 𝑥 → ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → ( 𝑧 ∈ On → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) ) ) |
99 |
98
|
imp |
⊢ ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → ( 𝑧 ∈ On → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) ) |
100 |
99
|
adantld |
⊢ ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( 𝑧 ∈ On → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) ) ) |
101 |
100
|
imp32 |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ∧ 𝑧 ∈ On ) ) → ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ) |
102 |
|
simplrr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ∧ 𝑧 ∈ On ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑧 ∈ On ) |
103 |
|
onelon |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ On ) |
104 |
103 30
|
sylan2 |
⊢ ( ( 𝐵 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) ) → ( 𝐵 +o 𝑦 ) ∈ On ) |
105 |
104
|
exp32 |
⊢ ( 𝐵 ∈ On → ( 𝑥 ∈ On → ( 𝑦 ∈ 𝑥 → ( 𝐵 +o 𝑦 ) ∈ On ) ) ) |
106 |
105
|
com12 |
⊢ ( 𝑥 ∈ On → ( 𝐵 ∈ On → ( 𝑦 ∈ 𝑥 → ( 𝐵 +o 𝑦 ) ∈ On ) ) ) |
107 |
106
|
imp31 |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝐵 +o 𝑦 ) ∈ On ) |
108 |
107
|
ad4ant24 |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ∧ 𝑧 ∈ On ) ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝐵 +o 𝑦 ) ∈ On ) |
109 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ∧ 𝑧 ∈ On ) → 𝐴 ∈ On ) |
110 |
109
|
ad2antlr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ∧ 𝑧 ∈ On ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝐴 ∈ On ) |
111 |
|
oaword |
⊢ ( ( 𝑧 ∈ On ∧ ( 𝐵 +o 𝑦 ) ∈ On ∧ 𝐴 ∈ On ) → ( 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ↔ ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) |
112 |
102 108 110 111
|
syl3anc |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ∧ 𝑧 ∈ On ) ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ↔ ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) |
113 |
112
|
rexbidva |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ∧ 𝑧 ∈ On ) ) → ( ∃ 𝑦 ∈ 𝑥 𝑧 ⊆ ( 𝐵 +o 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) |
114 |
101 113
|
mpbid |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ∧ 𝑧 ∈ On ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
115 |
114
|
exp32 |
⊢ ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( 𝑧 ∈ On → ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) ) |
116 |
61 115
|
mpdd |
⊢ ( ( Lim 𝑥 ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) |
117 |
116
|
exp32 |
⊢ ( Lim 𝑥 → ( 𝑥 ∈ On → ( 𝐵 ∈ On → ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) ) ) |
118 |
54 117
|
mpd |
⊢ ( Lim 𝑥 → ( 𝐵 ∈ On → ( ( 𝐴 ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) ) |
119 |
118
|
exp4a |
⊢ ( Lim 𝑥 → ( 𝐵 ∈ On → ( 𝐴 ∈ On → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) ) ) |
120 |
119
|
imp31 |
⊢ ( ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) ∧ 𝐴 ∈ On ) → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) |
121 |
120
|
ralrimiv |
⊢ ( ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) ∧ 𝐴 ∈ On ) → ∀ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
122 |
|
iunss2 |
⊢ ( ∀ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∃ 𝑦 ∈ 𝑥 ( 𝐴 +o 𝑧 ) ⊆ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ⊆ ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
123 |
121 122
|
syl |
⊢ ( ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) ∧ 𝐴 ∈ On ) → ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ⊆ ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
124 |
123
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ ( Lim 𝑥 ∧ 𝐵 ∈ On ) ) → ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ⊆ ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
125 |
|
oaordi |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑦 ∈ 𝑥 → ( 𝐵 +o 𝑦 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
126 |
125
|
anim1d |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝑦 ∈ 𝑥 ∧ 𝑤 ∈ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) → ( ( 𝐵 +o 𝑦 ) ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑤 ∈ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) ) |
127 |
|
oveq2 |
⊢ ( 𝑧 = ( 𝐵 +o 𝑦 ) → ( 𝐴 +o 𝑧 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
128 |
127
|
eleq2d |
⊢ ( 𝑧 = ( 𝐵 +o 𝑦 ) → ( 𝑤 ∈ ( 𝐴 +o 𝑧 ) ↔ 𝑤 ∈ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) ) |
129 |
128
|
rspcev |
⊢ ( ( ( 𝐵 +o 𝑦 ) ∈ ( 𝐵 +o 𝑥 ) ∧ 𝑤 ∈ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) 𝑤 ∈ ( 𝐴 +o 𝑧 ) ) |
130 |
126 129
|
syl6 |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝑦 ∈ 𝑥 ∧ 𝑤 ∈ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) 𝑤 ∈ ( 𝐴 +o 𝑧 ) ) ) |
131 |
130
|
expd |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑦 ∈ 𝑥 → ( 𝑤 ∈ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) 𝑤 ∈ ( 𝐴 +o 𝑧 ) ) ) ) |
132 |
131
|
rexlimdv |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( ∃ 𝑦 ∈ 𝑥 𝑤 ∈ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) 𝑤 ∈ ( 𝐴 +o 𝑧 ) ) ) |
133 |
|
eliun |
⊢ ( 𝑤 ∈ ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ↔ ∃ 𝑦 ∈ 𝑥 𝑤 ∈ ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
134 |
|
eliun |
⊢ ( 𝑤 ∈ ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ↔ ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) 𝑤 ∈ ( 𝐴 +o 𝑧 ) ) |
135 |
132 133 134
|
3imtr4g |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑤 ∈ ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → 𝑤 ∈ ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ) ) |
136 |
135
|
ssrdv |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ⊆ ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ) |
137 |
54 136
|
sylan |
⊢ ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) → ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ⊆ ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ) |
138 |
137
|
adantl |
⊢ ( ( 𝐴 ∈ On ∧ ( Lim 𝑥 ∧ 𝐵 ∈ On ) ) → ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ⊆ ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) ) |
139 |
124 138
|
eqssd |
⊢ ( ( 𝐴 ∈ On ∧ ( Lim 𝑥 ∧ 𝐵 ∈ On ) ) → ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 +o 𝑧 ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
140 |
52 139
|
eqtrd |
⊢ ( ( 𝐴 ∈ On ∧ ( Lim 𝑥 ∧ 𝐵 ∈ On ) ) → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
141 |
140
|
an12s |
⊢ ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
142 |
141
|
adantr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) → ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑦 ∈ 𝑥 ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) |
143 |
39 45 142
|
3eqtr4d |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) ) → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) ) |
144 |
143
|
exp31 |
⊢ ( Lim 𝑥 → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 +o 𝐵 ) +o 𝑦 ) = ( 𝐴 +o ( 𝐵 +o 𝑦 ) ) → ( ( 𝐴 +o 𝐵 ) +o 𝑥 ) = ( 𝐴 +o ( 𝐵 +o 𝑥 ) ) ) ) ) |
145 |
4 8 12 16 23 37 144
|
tfinds3 |
⊢ ( 𝐶 ∈ On → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) +o 𝐶 ) = ( 𝐴 +o ( 𝐵 +o 𝐶 ) ) ) ) |
146 |
145
|
com12 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐶 ∈ On → ( ( 𝐴 +o 𝐵 ) +o 𝐶 ) = ( 𝐴 +o ( 𝐵 +o 𝐶 ) ) ) ) |
147 |
146
|
3impia |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +o 𝐵 ) +o 𝐶 ) = ( 𝐴 +o ( 𝐵 +o 𝐶 ) ) ) |