Step | Hyp | Ref
| Expression |
1 | | elvv 5706 |
. . 3
⊢ (𝐺 ∈ (V × V) ↔
∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩) |
2 | | funsndifnop.g |
. . . . . 6
⊢ 𝐺 = {⟨𝐴, 𝐵⟩} |
3 | | funsndifnop.a |
. . . . . . . 8
⊢ 𝐴 ∈ V |
4 | | funsndifnop.b |
. . . . . . . 8
⊢ 𝐵 ∈ V |
5 | 3, 4 | funsn 6554 |
. . . . . . 7
⊢ Fun
{⟨𝐴, 𝐵⟩} |
6 | | funeq 6521 |
. . . . . . 7
⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → (Fun 𝐺 ↔ Fun {⟨𝐴, 𝐵⟩})) |
7 | 5, 6 | mpbiri 257 |
. . . . . 6
⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → Fun 𝐺) |
8 | 2, 7 | ax-mp 5 |
. . . . 5
⊢ Fun 𝐺 |
9 | | funeq 6521 |
. . . . . . 7
⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ Fun ⟨𝑥, 𝑦⟩)) |
10 | | vex 3449 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
11 | | vex 3449 |
. . . . . . . 8
⊢ 𝑦 ∈ V |
12 | 10, 11 | funop 7095 |
. . . . . . 7
⊢ (Fun
⟨𝑥, 𝑦⟩ ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩})) |
13 | 9, 12 | bitrdi 286 |
. . . . . 6
⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}))) |
14 | | eqeq2 2748 |
. . . . . . . . . . 11
⊢
(⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ ↔ 𝐺 = {⟨𝑎, 𝑎⟩})) |
15 | | eqeq1 2740 |
. . . . . . . . . . . . 13
⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩})) |
16 | | opex 5421 |
. . . . . . . . . . . . . . 15
⊢
⟨𝐴, 𝐵⟩ ∈ V |
17 | 16 | sneqr 4798 |
. . . . . . . . . . . . . 14
⊢
({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩) |
18 | 3, 4 | opth 5433 |
. . . . . . . . . . . . . . 15
⊢
(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ ↔ (𝐴 = 𝑎 ∧ 𝐵 = 𝑎)) |
19 | | eqtr3 2762 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑎) → 𝐴 = 𝐵) |
20 | 19 | a1d 25 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑎) → (𝑥 = {𝑎} → 𝐴 = 𝐵)) |
21 | 18, 20 | sylbi 216 |
. . . . . . . . . . . . . 14
⊢
(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵)) |
22 | 17, 21 | syl 17 |
. . . . . . . . . . . . 13
⊢
({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵)) |
23 | 15, 22 | syl6bi 252 |
. . . . . . . . . . . 12
⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))) |
24 | 2, 23 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (𝐺 = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵)) |
25 | 14, 24 | syl6bi 252 |
. . . . . . . . . 10
⊢
(⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵))) |
26 | 25 | com23 86 |
. . . . . . . . 9
⊢
(⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))) |
27 | 26 | impcom 408 |
. . . . . . . 8
⊢ ((𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)) |
28 | 27 | exlimiv 1933 |
. . . . . . 7
⊢
(∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)) |
29 | 28 | com12 32 |
. . . . . 6
⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → 𝐴 = 𝐵)) |
30 | 13, 29 | sylbid 239 |
. . . . 5
⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 → 𝐴 = 𝐵)) |
31 | 8, 30 | mpi 20 |
. . . 4
⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵) |
32 | 31 | exlimivv 1935 |
. . 3
⊢
(∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵) |
33 | 1, 32 | sylbi 216 |
. 2
⊢ (𝐺 ∈ (V × V) →
𝐴 = 𝐵) |
34 | 33 | necon3ai 2968 |
1
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V)) |