Step | Hyp | Ref
| Expression |
1 | | euabsn2 3439 |
. . 3
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧}) |
2 | | eleq2 2101 |
. . . . . 6
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧})) |
3 | | abid 2028 |
. . . . . 6
⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
4 | | velsn 3392 |
. . . . . 6
⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧) |
5 | 2, 3, 4 | 3bitr3g 211 |
. . . . 5
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ 𝑥 = 𝑧)) |
6 | | nfre1 2365 |
. . . . . . . . 9
⊢
Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑥 = 𝐵 |
7 | 6 | nfab 2182 |
. . . . . . . 8
⊢
Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} |
8 | 7 | nfeq1 2187 |
. . . . . . 7
⊢
Ⅎ𝑦{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} |
9 | | eusvobj1.1 |
. . . . . . . . 9
⊢ 𝐵 ∈ V |
10 | 9 | elabrex 5397 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵}) |
11 | | eleq2 2101 |
. . . . . . . . 9
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧})) |
12 | 9 | elsn 3391 |
. . . . . . . . . 10
⊢ (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧) |
13 | | eqcom 2042 |
. . . . . . . . . 10
⊢ (𝐵 = 𝑧 ↔ 𝑧 = 𝐵) |
14 | 12, 13 | bitri 173 |
. . . . . . . . 9
⊢ (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵) |
15 | 11, 14 | syl6bb 185 |
. . . . . . . 8
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵)) |
16 | 10, 15 | syl5ib 143 |
. . . . . . 7
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦 ∈ 𝐴 → 𝑧 = 𝐵)) |
17 | 8, 16 | ralrimi 2390 |
. . . . . 6
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) |
18 | | eqeq1 2046 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵)) |
19 | 18 | ralbidv 2326 |
. . . . . 6
⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵)) |
20 | 17, 19 | syl5ibrcom 146 |
. . . . 5
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
21 | 5, 20 | sylbid 139 |
. . . 4
⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
22 | 21 | exlimiv 1489 |
. . 3
⊢
(∃𝑧{𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
23 | 1, 22 | sylbi 114 |
. 2
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
24 | | euex 1930 |
. . 3
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
25 | | rexm 3320 |
. . . 4
⊢
(∃𝑦 ∈
𝐴 𝑥 = 𝐵 → ∃𝑦 𝑦 ∈ 𝐴) |
26 | 25 | exlimiv 1489 |
. . 3
⊢
(∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 𝑦 ∈ 𝐴) |
27 | | r19.2m 3309 |
. . . 4
⊢
((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵) |
28 | 27 | ex 108 |
. . 3
⊢
(∃𝑦 𝑦 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
29 | 24, 26, 28 | 3syl 17 |
. 2
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) |
30 | 23, 29 | impbid 120 |
1
⊢
(∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) |