Theorem prprc2 4734

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 4700	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2		prprc1 4733	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
3	1, 2	eqtrid 2812	1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  {cpr 4593

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3909  df-un 3911  df-nul 4287  df-sn 4592  df-pr 4594

This theorem is used by:  tpprceq3  4774  elpreqprlem  4833  prexOLD  5416  prfi  9290  indislem  23207  1to2vfriswmgr  30701  prssad  32946  indispconn  35763  bj-prmoore  37814  elsprel  48282