Theorem prprc2 4765

Description: A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)

Assertion

Ref	Expression
prprc2	⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

Proof of Theorem prprc2

Step	Hyp	Ref	Expression
1		prcom 4731	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2		prprc1 4764	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
3	1, 2	eqtrid 2778	1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468  {csn 4623  {cpr 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-nul 4318  df-sn 4624  df-pr 4626

This theorem is referenced by:  tpprceq3  4802  elpreqprlem  4861  prex  5425  indislem  22853  1to2vfriswmgr  30036  indispconn  34752  bj-prmoore  36502  elsprel  46697