Theorem opexOLD 5437

Description: Obsolete version of opex 5436 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 4831	. 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
2		prex 5400	. . 3 ⊢ {{𝐴}, {𝐴, 𝐵}} ∈ V
3		0ex 5262	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4534	. 2 ⊢ if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) ∈ V
5	1, 4	eqeltri 2861	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ V

Syntax hints:   ∧ wa 400   ∈ wcel 2145  Vcvv 3457  ∅c0 4288  ifcif 4483  {csn 4585  {cpr 4587  ⟨cop 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592

This theorem is referenced by: (None)