Theorem prprc2 4726

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 4692	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2		prprc1 4725	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
3	1, 2	eqtrid 2776	1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444  {csn 4585  {cpr 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-dif 3914  df-un 3916  df-nul 4293  df-sn 4586  df-pr 4588

This theorem is referenced by:  tpprceq3  4764  elpreqprlem  4826  prex  5387  prfi  9250  indislem  22863  1to2vfriswmgr  30181  prssad  32431  indispconn  35194  bj-prmoore  37076  elsprel  47449