Proof of Theorem preqsnd
Step | Hyp | Ref
| Expression |
1 | | preqsnd.1 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ V) |
2 | 1 | adantl 484 |
. . 3
⊢ ((𝐶 ∈ V ∧ 𝜑) → 𝐴 ∈ V) |
3 | | preqsnd.2 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ V) |
4 | 3 | adantl 484 |
. . 3
⊢ ((𝐶 ∈ V ∧ 𝜑) → 𝐵 ∈ V) |
5 | | simpl 485 |
. . 3
⊢ ((𝐶 ∈ V ∧ 𝜑) → 𝐶 ∈ V) |
6 | | dfsn2 4582 |
. . . . 5
⊢ {𝐶} = {𝐶, 𝐶} |
7 | 6 | eqeq2i 2836 |
. . . 4
⊢ ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶}) |
8 | | preq12bg 4786 |
. . . . 5
⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))) |
9 | | oridm 901 |
. . . . 5
⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
10 | 8, 9 | syl6bb 289 |
. . . 4
⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
11 | 7, 10 | syl5bb 285 |
. . 3
⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
12 | 2, 4, 5, 5, 11 | syl22anc 836 |
. 2
⊢ ((𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
13 | | snprc 4655 |
. . . . . . 7
⊢ (¬
𝐶 ∈ V ↔ {𝐶} = ∅) |
14 | 13 | biimpi 218 |
. . . . . 6
⊢ (¬
𝐶 ∈ V → {𝐶} = ∅) |
15 | 14 | adantr 483 |
. . . . 5
⊢ ((¬
𝐶 ∈ V ∧ 𝜑) → {𝐶} = ∅) |
16 | 15 | eqeq2d 2834 |
. . . 4
⊢ ((¬
𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = ∅)) |
17 | | prnzg 4715 |
. . . . . . 7
⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅) |
18 | | eqneqall 3029 |
. . . . . . 7
⊢ ({𝐴, 𝐵} = ∅ → ({𝐴, 𝐵} ≠ ∅ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
19 | 17, 18 | syl5com 31 |
. . . . . 6
⊢ (𝐴 ∈ V → ({𝐴, 𝐵} = ∅ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
20 | 1, 19 | syl 17 |
. . . . 5
⊢ (𝜑 → ({𝐴, 𝐵} = ∅ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
21 | 20 | adantl 484 |
. . . 4
⊢ ((¬
𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = ∅ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
22 | 16, 21 | sylbid 242 |
. . 3
⊢ ((¬
𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
23 | | eleq1 2902 |
. . . . . . . . . 10
⊢ (𝐶 = 𝐴 → (𝐶 ∈ V ↔ 𝐴 ∈ V)) |
24 | 23 | eqcoms 2831 |
. . . . . . . . 9
⊢ (𝐴 = 𝐶 → (𝐶 ∈ V ↔ 𝐴 ∈ V)) |
25 | 24 | notbid 320 |
. . . . . . . 8
⊢ (𝐴 = 𝐶 → (¬ 𝐶 ∈ V ↔ ¬ 𝐴 ∈ V)) |
26 | | pm2.21 123 |
. . . . . . . 8
⊢ (¬
𝐴 ∈ V → (𝐴 ∈ V → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶}))) |
27 | 25, 26 | syl6bi 255 |
. . . . . . 7
⊢ (𝐴 = 𝐶 → (¬ 𝐶 ∈ V → (𝐴 ∈ V → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))) |
28 | 27 | com13 88 |
. . . . . 6
⊢ (𝐴 ∈ V → (¬ 𝐶 ∈ V → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))) |
29 | 1, 28 | syl 17 |
. . . . 5
⊢ (𝜑 → (¬ 𝐶 ∈ V → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))) |
30 | 29 | impcom 410 |
. . . 4
⊢ ((¬
𝐶 ∈ V ∧ 𝜑) → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶}))) |
31 | 30 | impd 413 |
. . 3
⊢ ((¬
𝐶 ∈ V ∧ 𝜑) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶})) |
32 | 22, 31 | impbid 214 |
. 2
⊢ ((¬
𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
33 | 12, 32 | pm2.61ian 810 |
1
⊢ (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |