Theorem prprc2 4712

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 4678	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2		prprc1 4711	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
3	1, 2	eqtrid 2789	1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3441  {csn 4571  {cpr 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443  df-dif 3900  df-un 3902  df-nul 4268  df-sn 4572  df-pr 4574

This theorem is referenced by:  tpprceq3  4749  elpreqprlem  4808  prex  5370  indislem  22222  1to2vfriswmgr  28752  indispconn  33301  bj-prmoore  35342  elsprel  45179