Theorem opexOLD 5438

Description: Obsolete version of opex 5437 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 4840	. 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
2		prex 5401	. . 3 ⊢ {{𝐴}, {𝐴, 𝐵}} ∈ V
3		0ex 5265	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4536	. 2 ⊢ if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) ∈ V
5	1, 4	eqeltri 2860	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ V

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  ifcif 4485  {csn 4588  {cpr 4590  ⟨cop 4838

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839

This theorem is used by: (None)