Theorem opexOLD 5451

Description: Obsolete version of opex 5450 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 5414	. . 3 ⊢ {{𝐴}, {𝐴, 𝐵}} ∈ V
3		0ex 5275	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4543	. 2 ⊢ if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) ∈ V
5	1, 4	eqeltri 2862	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ V

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2146  Vcvv 3458  ∅c0 4289  ifcif 4492  {csn 4594  {cpr 4596  ⟨cop 4600

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601

This theorem is used by: (None)