Theorem opexOLD 5411

Description: Obsolete version of opex 5410 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 4808	. 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
2		prex 5374	. . 3 ⊢ {{𝐴}, {𝐴, 𝐵}} ∈ V
3		0ex 5236	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4512	. 2 ⊢ if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) ∈ V
5	1, 4	eqeltri 2836	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ V

Syntax hints:   ∧ wa 396   ∈ wcel 2119  Vcvv 3432  ∅c0 4268  ifcif 4461  {csn 4562  {cpr 4564  ⟨cop 4568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569

This theorem is referenced by: (None)