Theorem prexOLD 5394

Description: Obsolete version of prex 5389 as of 6-Mar-2026. (Contributed by NM, 15-Jul-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
prexOLD	⊢ {𝐴, 𝐵} ∈ V

Proof of Theorem prexOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq2 4687	. . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
2	1	eleq1d 2841	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3		zfpair2 5385	. . . . 5 ⊢ {𝑥, 𝑦} ∈ V
4	2, 3	vtoclg 3516	. . . 4 ⊢ (𝐵 ∈ V → {𝑥, 𝐵} ∈ V)
5		preq1 4686	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
6	5	eleq1d 2841	. . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V))
7	4, 6	imbitrid 246	. . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V → {𝐴, 𝐵} ∈ V))
8	7	vtocleg 3515	. 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ V → {𝐴, 𝐵} ∈ V))
9		prprc1 4718	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
10		snexOLD 5393	. . 3 ⊢ {𝐵} ∈ V
11	9, 10	eqeltrdi 2864	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} ∈ V)
12		prprc2 4719	. . 3 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
13		snexOLD 5393	. . 3 ⊢ {𝐴} ∈ V
14	12, 13	eqeltrdi 2864	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} ∈ V)
15	8, 11, 14	pm2.61nii 185	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1554   ∈ wcel 2136  Vcvv 3448  {csn 4576  {cpr 4578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-v 3450  df-dif 3902  df-un 3904  df-nul 4281  df-sn 4577  df-pr 4579

This theorem is referenced by: (None)