Step | Hyp | Ref
| Expression |
1 | | reldom 7847 |
. . . . 5
⊢ Rel
≼ |
2 | 1 | brrelex2i 5083 |
. . . 4
⊢ (ω
≼ 𝐴 → 𝐴 ∈ V) |
3 | | cdadom3 8893 |
. . . 4
⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ≼ (𝐴 +𝑐 𝐴)) |
4 | 2, 2, 3 | syl2anc 691 |
. . 3
⊢ (ω
≼ 𝐴 → 𝐴 ≼ (𝐴 +𝑐 𝐴)) |
5 | | domtr 7895 |
. . 3
⊢ ((ω
≼ 𝐴 ∧ 𝐴 ≼ (𝐴 +𝑐 𝐴)) → ω ≼ (𝐴 +𝑐 𝐴)) |
6 | 4, 5 | mpdan 699 |
. 2
⊢ (ω
≼ 𝐴 → ω
≼ (𝐴
+𝑐 𝐴)) |
7 | | infn0 8107 |
. . . 4
⊢ (ω
≼ (𝐴
+𝑐 𝐴)
→ (𝐴
+𝑐 𝐴)
≠ ∅) |
8 | | cdafn 8874 |
. . . . . . . 8
⊢
+𝑐 Fn (V × V) |
9 | | fndm 5904 |
. . . . . . . 8
⊢ (
+𝑐 Fn (V × V) → dom +𝑐 = (V
× V)) |
10 | 8, 9 | ax-mp 5 |
. . . . . . 7
⊢ dom
+𝑐 = (V × V) |
11 | 10 | ndmov 6716 |
. . . . . 6
⊢ (¬
(𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ∅) |
12 | 11 | necon1ai 2809 |
. . . . 5
⊢ ((𝐴 +𝑐 𝐴) ≠ ∅ → (𝐴 ∈ V ∧ 𝐴 ∈ V)) |
13 | 12 | simpld 474 |
. . . 4
⊢ ((𝐴 +𝑐 𝐴) ≠ ∅ → 𝐴 ∈ V) |
14 | 7, 13 | syl 17 |
. . 3
⊢ (ω
≼ (𝐴
+𝑐 𝐴)
→ 𝐴 ∈
V) |
15 | | ovex 6577 |
. . . . 5
⊢ (𝐴 +𝑐 𝐴) ∈ V |
16 | 15 | domen 7854 |
. . . 4
⊢ (ω
≼ (𝐴
+𝑐 𝐴)
↔ ∃𝑥(ω
≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) |
17 | | indi 3832 |
. . . . . . . . 9
⊢ (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) =
((𝑥 ∩ (𝐴 × {∅})) ∪
(𝑥 ∩ (𝐴 ×
{1𝑜}))) |
18 | | simprr 792 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧ (ω ≈
𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ (𝐴 +𝑐 𝐴)) |
19 | | simpl 472 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ V ∧ (ω ≈
𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝐴 ∈ V) |
20 | | cdaval 8875 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 ×
{1𝑜}))) |
21 | 19, 19, 20 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧ (ω ≈
𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 ×
{1𝑜}))) |
22 | 18, 21 | sseqtrd 3604 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ V ∧ (ω ≈
𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 ×
{1𝑜}))) |
23 | | df-ss 3554 |
. . . . . . . . . 10
⊢ (𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})) ↔
(𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 ×
{1𝑜}))) = 𝑥) |
24 | 22, 23 | sylib 207 |
. . . . . . . . 9
⊢ ((𝐴 ∈ V ∧ (ω ≈
𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥) |
25 | 17, 24 | syl5eqr 2658 |
. . . . . . . 8
⊢ ((𝐴 ∈ V ∧ (ω ≈
𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) = 𝑥) |
26 | | ensym 7891 |
. . . . . . . . 9
⊢ (ω
≈ 𝑥 → 𝑥 ≈
ω) |
27 | 26 | ad2antrl 760 |
. . . . . . . 8
⊢ ((𝐴 ∈ V ∧ (ω ≈
𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ≈ ω) |
28 | 25, 27 | eqbrtrd 4605 |
. . . . . . 7
⊢ ((𝐴 ∈ V ∧ (ω ≈
𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈
ω) |
29 | 28 | ex 449 |
. . . . . 6
⊢ (𝐴 ∈ V → ((ω
≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈
ω)) |
30 | | cdainflem 8896 |
. . . . . . 7
⊢ (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈
ω → ((𝑥 ∩
(𝐴 × {∅}))
≈ ω ∨ (𝑥
∩ (𝐴 ×
{1𝑜})) ≈ ω)) |
31 | | snex 4835 |
. . . . . . . . . . . 12
⊢ {∅}
∈ V |
32 | | xpexg 6858 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ V ∧ {∅} ∈
V) → (𝐴 ×
{∅}) ∈ V) |
33 | 31, 32 | mpan2 703 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ V → (𝐴 × {∅}) ∈
V) |
34 | | inss2 3796 |
. . . . . . . . . . 11
⊢ (𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 ×
{∅}) |
35 | | ssdomg 7887 |
. . . . . . . . . . 11
⊢ ((𝐴 × {∅}) ∈ V
→ ((𝑥 ∩ (𝐴 × {∅})) ⊆
(𝐴 × {∅})
→ (𝑥 ∩ (𝐴 × {∅})) ≼
(𝐴 ×
{∅}))) |
36 | 33, 34, 35 | mpisyl 21 |
. . . . . . . . . 10
⊢ (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 ×
{∅})) |
37 | | 0ex 4718 |
. . . . . . . . . . 11
⊢ ∅
∈ V |
38 | | xpsneng 7930 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧ ∅ ∈
V) → (𝐴 ×
{∅}) ≈ 𝐴) |
39 | 37, 38 | mpan2 703 |
. . . . . . . . . 10
⊢ (𝐴 ∈ V → (𝐴 × {∅}) ≈
𝐴) |
40 | | domentr 7901 |
. . . . . . . . . 10
⊢ (((𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈
𝐴) → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴) |
41 | 36, 39, 40 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴) |
42 | | domen1 7987 |
. . . . . . . . 9
⊢ ((𝑥 ∩ (𝐴 × {∅})) ≈ ω →
((𝑥 ∩ (𝐴 × {∅})) ≼
𝐴 ↔ ω ≼
𝐴)) |
43 | 41, 42 | syl5ibcom 234 |
. . . . . . . 8
⊢ (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω →
ω ≼ 𝐴)) |
44 | | snex 4835 |
. . . . . . . . . . . 12
⊢
{1𝑜} ∈ V |
45 | | xpexg 6858 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ V ∧
{1𝑜} ∈ V) → (𝐴 × {1𝑜}) ∈
V) |
46 | 44, 45 | mpan2 703 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ V → (𝐴 ×
{1𝑜}) ∈ V) |
47 | | inss2 3796 |
. . . . . . . . . . 11
⊢ (𝑥 ∩ (𝐴 × {1𝑜})) ⊆
(𝐴 ×
{1𝑜}) |
48 | | ssdomg 7887 |
. . . . . . . . . . 11
⊢ ((𝐴 ×
{1𝑜}) ∈ V → ((𝑥 ∩ (𝐴 × {1𝑜})) ⊆
(𝐴 ×
{1𝑜}) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼
(𝐴 ×
{1𝑜}))) |
49 | 46, 47, 48 | mpisyl 21 |
. . . . . . . . . 10
⊢ (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼
(𝐴 ×
{1𝑜})) |
50 | | 1on 7454 |
. . . . . . . . . . 11
⊢
1𝑜 ∈ On |
51 | | xpsneng 7930 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧
1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈
𝐴) |
52 | 50, 51 | mpan2 703 |
. . . . . . . . . 10
⊢ (𝐴 ∈ V → (𝐴 ×
{1𝑜}) ≈ 𝐴) |
53 | | domentr 7901 |
. . . . . . . . . 10
⊢ (((𝑥 ∩ (𝐴 × {1𝑜})) ≼
(𝐴 ×
{1𝑜}) ∧ (𝐴 × {1𝑜}) ≈
𝐴) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼
𝐴) |
54 | 49, 52, 53 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼
𝐴) |
55 | | domen1 7987 |
. . . . . . . . 9
⊢ ((𝑥 ∩ (𝐴 × {1𝑜})) ≈
ω → ((𝑥 ∩
(𝐴 ×
{1𝑜})) ≼ 𝐴 ↔ ω ≼ 𝐴)) |
56 | 54, 55 | syl5ibcom 234 |
. . . . . . . 8
⊢ (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {1𝑜})) ≈
ω → ω ≼ 𝐴)) |
57 | 43, 56 | jaod 394 |
. . . . . . 7
⊢ (𝐴 ∈ V → (((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨
(𝑥 ∩ (𝐴 × {1𝑜})) ≈
ω) → ω ≼ 𝐴)) |
58 | 30, 57 | syl5 33 |
. . . . . 6
⊢ (𝐴 ∈ V → (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈
ω → ω ≼ 𝐴)) |
59 | 29, 58 | syld 46 |
. . . . 5
⊢ (𝐴 ∈ V → ((ω
≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴)) |
60 | 59 | exlimdv 1848 |
. . . 4
⊢ (𝐴 ∈ V → (∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴)) |
61 | 16, 60 | syl5bi 231 |
. . 3
⊢ (𝐴 ∈ V → (ω
≼ (𝐴
+𝑐 𝐴)
→ ω ≼ 𝐴)) |
62 | 14, 61 | mpcom 37 |
. 2
⊢ (ω
≼ (𝐴
+𝑐 𝐴)
→ ω ≼ 𝐴) |
63 | 6, 62 | impbii 198 |
1
⊢ (ω
≼ 𝐴 ↔ ω
≼ (𝐴
+𝑐 𝐴)) |