Proof of Theorem reusv2lem3
Step | Hyp | Ref
| Expression |
1 | | simpr 488 |
. . . 4
⊢
((∀𝑦 ∈
𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
2 | | nfv 1922 |
. . . . . 6
⊢
Ⅎ𝑥∀𝑦 ∈ 𝐴 𝐵 ∈ V |
3 | | nfeu1 2587 |
. . . . . 6
⊢
Ⅎ𝑥∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 |
4 | 2, 3 | nfan 1907 |
. . . . 5
⊢
Ⅎ𝑥(∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
5 | | euex 2576 |
. . . . . . . 8
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
6 | | rexn0 4422 |
. . . . . . . . 9
⊢
(∃𝑦 ∈
𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅) |
7 | 6 | exlimiv 1938 |
. . . . . . . 8
⊢
(∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅) |
8 | | r19.2z 4406 |
. . . . . . . . 9
⊢ ((𝐴 ≠ ∅ ∧
∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
9 | 8 | ex 416 |
. . . . . . . 8
⊢ (𝐴 ≠ ∅ →
(∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
10 | 5, 7, 9 | 3syl 18 |
. . . . . . 7
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
11 | 10 | adantl 485 |
. . . . . 6
⊢
((∀𝑦 ∈
𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
12 | | nfra1 3140 |
. . . . . . . 8
⊢
Ⅎ𝑦∀𝑦 ∈ 𝐴 𝐵 ∈ V |
13 | | nfre1 3225 |
. . . . . . . . 9
⊢
Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑥 = 𝐵 |
14 | 13 | nfeuw 2592 |
. . . . . . . 8
⊢
Ⅎ𝑦∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 |
15 | 12, 14 | nfan 1907 |
. . . . . . 7
⊢
Ⅎ𝑦(∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
16 | | rsp 3127 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 ∈
𝐴 𝐵 ∈ V → (𝑦 ∈ 𝐴 → 𝐵 ∈ V)) |
17 | 16 | impcom 411 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝐵 ∈ V) → 𝐵 ∈ V) |
18 | | isset 3421 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵) |
19 | 17, 18 | sylib 221 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝐵 ∈ V) → ∃𝑥 𝑥 = 𝐵) |
20 | 19 | adantrr 717 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → ∃𝑥 𝑥 = 𝐵) |
21 | | rspe 3223 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
22 | 21 | ex 416 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
23 | 22 | ancrd 555 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵))) |
24 | 23 | eximdv 1925 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ 𝐴 → (∃𝑥 𝑥 = 𝐵 → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵))) |
25 | 24 | imp 410 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝐴 ∧ ∃𝑥 𝑥 = 𝐵) → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)) |
26 | 20, 25 | syldan 594 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)) |
27 | | eupick 2634 |
. . . . . . . . . 10
⊢
((∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵)) |
28 | 1, 26, 27 | syl2an2 686 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵)) |
29 | 28 | ex 416 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵))) |
30 | 29 | com3l 89 |
. . . . . . 7
⊢
((∀𝑦 ∈
𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (𝑦 ∈ 𝐴 → 𝑥 = 𝐵))) |
31 | 15, 13, 30 | ralrimd 3139 |
. . . . . 6
⊢
((∀𝑦 ∈
𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
32 | 11, 31 | impbid 215 |
. . . . 5
⊢
((∀𝑦 ∈
𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
33 | 4, 32 | eubid 2586 |
. . . 4
⊢
((∀𝑦 ∈
𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
34 | 1, 33 | mpbird 260 |
. . 3
⊢
((∀𝑦 ∈
𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) |
35 | 34 | ex 416 |
. 2
⊢
(∀𝑦 ∈
𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
36 | | reusv2lem2 5292 |
. 2
⊢
(∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
37 | 35, 36 | impbid1 228 |
1
⊢
(∀𝑦 ∈
𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |