Theorem prprc2 4723

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 4689	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2		prprc1 4722	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
3	1, 2	eqtrid 2807	1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4046  df-un 4150  df-nul 4278  df-sn 4585  df-pr 4587

This theorem is used by:  tpprceq3  4763  elpreqprlem  4820  prexOLD  5401  prfi  9322  indislem  23327  1to2vfriswmgr  30889  prssad  33176  indispconn  36089  bj-prmoore  38144  elsprel  48665