Theorem altopex 32670

Description: Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)

Assertion

Ref	Expression
altopex	⊢ ⟪𝐴, 𝐵⟫ ∈ V

Proof of Theorem altopex

Step	Hyp	Ref	Expression
1		df-altop 32668	. 2 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2		prex 5141	. 2 ⊢ {{𝐴}, {𝐴, {𝐵}}} ∈ V
3	1, 2	eqeltri 2854	1 ⊢ ⟪𝐴, 𝐵⟫ ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3397  {csn 4397  {cpr 4399  ⟪caltop 32666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-v 3399  df-dif 3794  df-un 3796  df-nul 4141  df-sn 4398  df-pr 4400  df-altop 32668

This theorem is referenced by:  elaltxp  32685