Proof of Theorem ttukeylem1
Step | Hyp | Ref
| Expression |
1 | | elex 3185 |
. . 3
⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) |
2 | 1 | a1i 11 |
. 2
⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ V)) |
3 | | id 22 |
. . . . 5
⊢
((𝒫 𝐶 ∩
Fin) ⊆ 𝐴 →
(𝒫 𝐶 ∩ Fin)
⊆ 𝐴) |
4 | | ssun1 3738 |
. . . . . . . 8
⊢ ∪ 𝐴
⊆ (∪ 𝐴 ∪ 𝐵) |
5 | | undif1 3995 |
. . . . . . . 8
⊢ ((∪ 𝐴
∖ 𝐵) ∪ 𝐵) = (∪ 𝐴
∪ 𝐵) |
6 | 4, 5 | sseqtr4i 3601 |
. . . . . . 7
⊢ ∪ 𝐴
⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) |
7 | | fvex 6113 |
. . . . . . . . 9
⊢
(card‘(∪ 𝐴 ∖ 𝐵)) ∈ V |
8 | | ttukeylem.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:(card‘(∪
𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) |
9 | | f1ofo 6057 |
. . . . . . . . . 10
⊢ (𝐹:(card‘(∪ 𝐴
∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) → 𝐹:(card‘(∪
𝐴 ∖ 𝐵))–onto→(∪ 𝐴 ∖ 𝐵)) |
10 | 8, 9 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:(card‘(∪
𝐴 ∖ 𝐵))–onto→(∪ 𝐴 ∖ 𝐵)) |
11 | | fornex 7028 |
. . . . . . . . 9
⊢
((card‘(∪ 𝐴 ∖ 𝐵)) ∈ V → (𝐹:(card‘(∪
𝐴 ∖ 𝐵))–onto→(∪ 𝐴 ∖ 𝐵) → (∪ 𝐴 ∖ 𝐵) ∈ V)) |
12 | 7, 10, 11 | mpsyl 66 |
. . . . . . . 8
⊢ (𝜑 → (∪ 𝐴
∖ 𝐵) ∈
V) |
13 | | ttukeylem.2 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ 𝐴) |
14 | | unexg 6857 |
. . . . . . . 8
⊢ (((∪ 𝐴
∖ 𝐵) ∈ V ∧
𝐵 ∈ 𝐴) → ((∪
𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) |
15 | 12, 13, 14 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → ((∪ 𝐴
∖ 𝐵) ∪ 𝐵) ∈ V) |
16 | | ssexg 4732 |
. . . . . . 7
⊢ ((∪ 𝐴
⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → ∪ 𝐴
∈ V) |
17 | 6, 15, 16 | sylancr 694 |
. . . . . 6
⊢ (𝜑 → ∪ 𝐴
∈ V) |
18 | | uniexb 6866 |
. . . . . 6
⊢ (𝐴 ∈ V ↔ ∪ 𝐴
∈ V) |
19 | 17, 18 | sylibr 223 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ V) |
20 | | ssexg 4732 |
. . . . 5
⊢
(((𝒫 𝐶 ∩
Fin) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝒫
𝐶 ∩ Fin) ∈
V) |
21 | 3, 19, 20 | syl2anr 494 |
. . . 4
⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → (𝒫 𝐶 ∩ Fin) ∈
V) |
22 | | infpwfidom 8734 |
. . . 4
⊢
((𝒫 𝐶 ∩
Fin) ∈ V → 𝐶
≼ (𝒫 𝐶 ∩
Fin)) |
23 | | reldom 7847 |
. . . . 5
⊢ Rel
≼ |
24 | 23 | brrelexi 5082 |
. . . 4
⊢ (𝐶 ≼ (𝒫 𝐶 ∩ Fin) → 𝐶 ∈ V) |
25 | 21, 22, 24 | 3syl 18 |
. . 3
⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → 𝐶 ∈ V) |
26 | 25 | ex 449 |
. 2
⊢ (𝜑 → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → 𝐶 ∈ V)) |
27 | | ttukeylem.3 |
. . 3
⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) |
28 | | eleq1 2676 |
. . . . 5
⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
29 | | pweq 4111 |
. . . . . . 7
⊢ (𝑥 = 𝐶 → 𝒫 𝑥 = 𝒫 𝐶) |
30 | 29 | ineq1d 3775 |
. . . . . 6
⊢ (𝑥 = 𝐶 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝐶 ∩ Fin)) |
31 | 30 | sseq1d 3595 |
. . . . 5
⊢ (𝑥 = 𝐶 → ((𝒫 𝑥 ∩ Fin) ⊆ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)) |
32 | 28, 31 | bibi12d 334 |
. . . 4
⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) ↔ (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))) |
33 | 32 | spcgv 3266 |
. . 3
⊢ (𝐶 ∈ V → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))) |
34 | 27, 33 | syl5com 31 |
. 2
⊢ (𝜑 → (𝐶 ∈ V → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))) |
35 | 2, 26, 34 | pm5.21ndd 368 |
1
⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)) |