Theorem prprc2 4725

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 4691	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2		prprc1 4724	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
3	1, 2	eqtrid 2784	1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582  {cpr 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-un 3908  df-nul 4288  df-sn 4583  df-pr 4585

This theorem is referenced by:  tpprceq3  4762  elpreqprlem  4824  prexOLD  5389  prfi  9236  indislem  22956  1to2vfriswmgr  30366  prssad  32616  indispconn  35450  bj-prmoore  37368  elsprel  47835