Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑐 = 𝑑 → ( 𝐴 +no 𝑐 ) = ( 𝐴 +no 𝑑 ) ) |
2 |
|
oveq2 |
⊢ ( 𝑐 = 𝑑 → ( 𝐵 +no 𝑐 ) = ( 𝐵 +no 𝑑 ) ) |
3 |
1 2
|
sseq12d |
⊢ ( 𝑐 = 𝑑 → ( ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ↔ ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝑐 = 𝑑 → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ↔ ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) ) |
5 |
4
|
imbi2d |
⊢ ( 𝑐 = 𝑑 → ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ) ↔ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) ) ) |
6 |
|
oveq2 |
⊢ ( 𝑐 = 𝐶 → ( 𝐴 +no 𝑐 ) = ( 𝐴 +no 𝐶 ) ) |
7 |
|
oveq2 |
⊢ ( 𝑐 = 𝐶 → ( 𝐵 +no 𝑐 ) = ( 𝐵 +no 𝐶 ) ) |
8 |
6 7
|
sseq12d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ↔ ( 𝐴 +no 𝐶 ) ⊆ ( 𝐵 +no 𝐶 ) ) ) |
9 |
8
|
imbi2d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ↔ ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝐶 ) ⊆ ( 𝐵 +no 𝐶 ) ) ) ) |
10 |
9
|
imbi2d |
⊢ ( 𝑐 = 𝐶 → ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ) ↔ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝐶 ) ⊆ ( 𝐵 +no 𝐶 ) ) ) ) ) |
11 |
|
r19.21v |
⊢ ( ∀ 𝑑 ∈ 𝑐 ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) ↔ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ∀ 𝑑 ∈ 𝑐 ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) ) |
12 |
|
r19.21v |
⊢ ( ∀ 𝑑 ∈ 𝑐 ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ↔ ( 𝐴 ⊆ 𝐵 → ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) |
13 |
12
|
imbi2i |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ∀ 𝑑 ∈ 𝑐 ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) ↔ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) ) |
14 |
11 13
|
bitri |
⊢ ( ∀ 𝑑 ∈ 𝑐 ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) ↔ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑑 = 𝑤 → ( 𝐴 +no 𝑑 ) = ( 𝐴 +no 𝑤 ) ) |
16 |
|
oveq2 |
⊢ ( 𝑑 = 𝑤 → ( 𝐵 +no 𝑑 ) = ( 𝐵 +no 𝑤 ) ) |
17 |
15 16
|
sseq12d |
⊢ ( 𝑑 = 𝑤 → ( ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ↔ ( 𝐴 +no 𝑤 ) ⊆ ( 𝐵 +no 𝑤 ) ) ) |
18 |
17
|
rspccva |
⊢ ( ( ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ∧ 𝑤 ∈ 𝑐 ) → ( 𝐴 +no 𝑤 ) ⊆ ( 𝐵 +no 𝑤 ) ) |
19 |
18
|
ad4ant24 |
⊢ ( ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) ∧ 𝑤 ∈ 𝑐 ) → ( 𝐴 +no 𝑤 ) ⊆ ( 𝐵 +no 𝑤 ) ) |
20 |
|
simprrl |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) → ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ) |
21 |
|
oveq2 |
⊢ ( 𝑦 = 𝑤 → ( 𝐵 +no 𝑦 ) = ( 𝐵 +no 𝑤 ) ) |
22 |
21
|
eleq1d |
⊢ ( 𝑦 = 𝑤 → ( ( 𝐵 +no 𝑦 ) ∈ 𝑥 ↔ ( 𝐵 +no 𝑤 ) ∈ 𝑥 ) ) |
23 |
22
|
rspccva |
⊢ ( ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ 𝑤 ∈ 𝑐 ) → ( 𝐵 +no 𝑤 ) ∈ 𝑥 ) |
24 |
20 23
|
sylan |
⊢ ( ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) ∧ 𝑤 ∈ 𝑐 ) → ( 𝐵 +no 𝑤 ) ∈ 𝑥 ) |
25 |
|
simplrl |
⊢ ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ∈ On ) |
26 |
25
|
adantr |
⊢ ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → 𝐴 ∈ On ) |
27 |
26
|
adantr |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) → 𝐴 ∈ On ) |
28 |
27
|
adantr |
⊢ ( ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) ∧ 𝑤 ∈ 𝑐 ) → 𝐴 ∈ On ) |
29 |
|
simp-4l |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) → 𝑐 ∈ On ) |
30 |
|
onelon |
⊢ ( ( 𝑐 ∈ On ∧ 𝑤 ∈ 𝑐 ) → 𝑤 ∈ On ) |
31 |
29 30
|
sylan |
⊢ ( ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) ∧ 𝑤 ∈ 𝑐 ) → 𝑤 ∈ On ) |
32 |
|
naddcl |
⊢ ( ( 𝐴 ∈ On ∧ 𝑤 ∈ On ) → ( 𝐴 +no 𝑤 ) ∈ On ) |
33 |
28 31 32
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) ∧ 𝑤 ∈ 𝑐 ) → ( 𝐴 +no 𝑤 ) ∈ On ) |
34 |
|
simplrl |
⊢ ( ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) ∧ 𝑤 ∈ 𝑐 ) → 𝑥 ∈ On ) |
35 |
|
ontr2 |
⊢ ( ( ( 𝐴 +no 𝑤 ) ∈ On ∧ 𝑥 ∈ On ) → ( ( ( 𝐴 +no 𝑤 ) ⊆ ( 𝐵 +no 𝑤 ) ∧ ( 𝐵 +no 𝑤 ) ∈ 𝑥 ) → ( 𝐴 +no 𝑤 ) ∈ 𝑥 ) ) |
36 |
33 34 35
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) ∧ 𝑤 ∈ 𝑐 ) → ( ( ( 𝐴 +no 𝑤 ) ⊆ ( 𝐵 +no 𝑤 ) ∧ ( 𝐵 +no 𝑤 ) ∈ 𝑥 ) → ( 𝐴 +no 𝑤 ) ∈ 𝑥 ) ) |
37 |
19 24 36
|
mp2and |
⊢ ( ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) ∧ 𝑤 ∈ 𝑐 ) → ( 𝐴 +no 𝑤 ) ∈ 𝑥 ) |
38 |
37
|
ralrimiva |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) → ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ) |
39 |
|
simpllr |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) → 𝐴 ⊆ 𝐵 ) |
40 |
|
simprrr |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) → ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) |
41 |
|
ssralv |
⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 → ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) |
42 |
39 40 41
|
sylc |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) → ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) |
43 |
38 42
|
jca |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ ( 𝑥 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) → ( ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) |
44 |
43
|
expr |
⊢ ( ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ∧ 𝑥 ∈ On ) → ( ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) → ( ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) ) ) |
45 |
44
|
ss2rabdv |
⊢ ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → { 𝑥 ∈ On ∣ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ⊆ { 𝑥 ∈ On ∣ ( ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ) |
46 |
|
intss |
⊢ ( { 𝑥 ∈ On ∣ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ⊆ { 𝑥 ∈ On ∣ ( ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } → ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ⊆ ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ) |
47 |
45 46
|
syl |
⊢ ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ⊆ ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ) |
48 |
|
simplll |
⊢ ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → 𝑐 ∈ On ) |
49 |
|
naddov2 |
⊢ ( ( 𝐴 ∈ On ∧ 𝑐 ∈ On ) → ( 𝐴 +no 𝑐 ) = ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ) |
50 |
26 48 49
|
syl2anc |
⊢ ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → ( 𝐴 +no 𝑐 ) = ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑤 ∈ 𝑐 ( 𝐴 +no 𝑤 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ) |
51 |
|
simplrr |
⊢ ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐵 ∈ On ) |
52 |
51
|
adantr |
⊢ ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → 𝐵 ∈ On ) |
53 |
|
naddov2 |
⊢ ( ( 𝐵 ∈ On ∧ 𝑐 ∈ On ) → ( 𝐵 +no 𝑐 ) = ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ) |
54 |
52 48 53
|
syl2anc |
⊢ ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → ( 𝐵 +no 𝑐 ) = ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑦 ∈ 𝑐 ( 𝐵 +no 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( 𝑧 +no 𝑐 ) ∈ 𝑥 ) } ) |
55 |
47 50 54
|
3sstr4d |
⊢ ( ( ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝐴 ⊆ 𝐵 ) ∧ ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) |
56 |
55
|
exp31 |
⊢ ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ) ) |
57 |
56
|
a2d |
⊢ ( ( 𝑐 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( ( 𝐴 ⊆ 𝐵 → ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ) ) |
58 |
57
|
ex |
⊢ ( 𝑐 ∈ On → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 → ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ) ) ) |
59 |
58
|
a2d |
⊢ ( 𝑐 ∈ On → ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ∀ 𝑑 ∈ 𝑐 ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ) ) ) |
60 |
14 59
|
syl5bi |
⊢ ( 𝑐 ∈ On → ( ∀ 𝑑 ∈ 𝑐 ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑑 ) ⊆ ( 𝐵 +no 𝑑 ) ) ) → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝑐 ) ⊆ ( 𝐵 +no 𝑐 ) ) ) ) ) |
61 |
5 10 60
|
tfis3 |
⊢ ( 𝐶 ∈ On → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝐶 ) ⊆ ( 𝐵 +no 𝐶 ) ) ) ) |
62 |
61
|
com12 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐶 ∈ On → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝐶 ) ⊆ ( 𝐵 +no 𝐶 ) ) ) ) |
63 |
62
|
3impia |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +no 𝐶 ) ⊆ ( 𝐵 +no 𝐶 ) ) ) |