Theorem prexOLD 5413

Description: Obsolete version of prex 5408 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 4699	. . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
2	1	eleq1d 2847	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3		zfpair2 5404	. . . . 5 ⊢ {𝑥, 𝑦} ∈ V
4	2, 3	vtoclg 3521	. . . 4 ⊢ (𝐵 ∈ V → {𝑥, 𝐵} ∈ V)
5		preq1 4698	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
6	5	eleq1d 2847	. . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V))
7	4, 6	imbitrid 247	. . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V → {𝐴, 𝐵} ∈ V))
8	7	vtocleg 3520	. 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ V → {𝐴, 𝐵} ∈ V))
9		prprc1 4730	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
10		snexOLD 5412	. . 3 ⊢ {𝐵} ∈ V
11	9, 10	eqeltrdi 2870	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} ∈ V)
12		prprc2 4731	. . 3 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
13		snexOLD 5412	. . 3 ⊢ {𝐴} ∈ V
14	12, 13	eqeltrdi 2870	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} ∈ V)
15	8, 11, 14	pm2.61nii 186	1 ⊢ {𝐴, 𝐵} ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2142  Vcvv 3454  {csn 4588  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4589  df-pr 4591

This theorem is used by: (None)