Theorem altopex 32404

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 32402	. 2 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
2		prex 5038	. 2 ⊢ {{𝐴}, {𝐴, {𝐵}}} ∈ V
3	1, 2	eqeltri 2846	1 ⊢ ⟪𝐴, 𝐵⟫ ∈ V

Syntax hints:   ∈ wcel 2145  Vcvv 3351  {csn 4317  {cpr 4319  ⟪caltop 32400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4318  df-pr 4320  df-altop 32402

This theorem is referenced by:  elaltxp  32419