Theorem opexOLD 5420

Description: Obsolete version of opex 5419 as of 6-Mar-2026. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
opexOLD	⊢ ⟨𝐴, 𝐵⟩ ∈ V

Proof of Theorem opexOLD

Step	Hyp	Ref	Expression
1		dfopif 4828	. 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
2		prex 5384	. . 3 ⊢ {{𝐴}, {𝐴, 𝐵}} ∈ V
3		0ex 5254	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4532	. 2 ⊢ if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) ∈ V
5	1, 4	eqeltri 2833	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ V

Syntax hints:   ∧ wa 395   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ifcif 4481  {csn 4582  {cpr 4584  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by: (None)