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Theorem prprc1 4096
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)
Assertion
Ref Expression
prprc1  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )

Proof of Theorem prprc1
StepHypRef Expression
1 snprc 4050 . 2  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 uneq1 3614 . . 3  |-  ( { A }  =  (/)  ->  ( { A }  u.  { B } )  =  ( (/)  u.  { B } ) )
3 df-pr 3991 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 uncom 3611 . . . 4  |-  ( (/)  u. 
{ B } )  =  ( { B }  u.  (/) )
5 un0 3773 . . . 4  |-  ( { B }  u.  (/) )  =  { B }
64, 5eqtr2i 2484 . . 3  |-  { B }  =  ( (/)  u.  { B } )
72, 3, 63eqtr4g 2520 . 2  |-  ( { A }  =  (/)  ->  { A ,  B }  =  { B } )
81, 7sylbi 195 1  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    u. cun 3437   (/)c0 3748   {csn 3988   {cpr 3990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-dif 3442  df-un 3444  df-nul 3749  df-sn 3989  df-pr 3991
This theorem is referenced by:  prprc2  4097  prprc  4098  prex  4645  elprchashprn2  12278  usgraedgprv  23474
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