Theorem prprc2 4729

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 4695	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2		prprc1 4728	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
3	1, 2	eqtrid 2810	1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591

This theorem is used by:  tpprceq3  4769  elpreqprlem  4826  prexOLD  5408  prfi  9353  indislem  23373  1to2vfriswmgr  30937  prssad  33224  indispconn  36207  bj-prmoore  38306  elsprel  48830