Theorem prprc2 2453

Description: A proper class vanishes in an unordered pair.

Assertion

Ref	Expression
prprc2	|- (-. B e. V -> {A, B} = {A})

Proof of Theorem prprc2

Step	Hyp	Ref	Expression
1		prprc1 2452	. 2 |- (-. B e. V -> {B, A} = {A})
2		prcom 2447	. 2 |- {A, B} = {B, A}
3	1, 2	syl5eq 1519	1 |- (-. B e. V -> {A, B} = {A})

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  {cpr 2410

This theorem is referenced by:  prex 2781

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281  df-sn 2412  df-pr 2413

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