MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prprc1 Unicode version

Theorem prprc1 3874
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
prprc1  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )

Proof of Theorem prprc1
StepHypRef Expression
1 snprc 3831 . 2  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 uneq1 3454 . . 3  |-  ( { A }  =  (/)  ->  ( { A }  u.  { B } )  =  ( (/)  u.  { B } ) )
3 df-pr 3781 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 uncom 3451 . . . 4  |-  ( (/)  u. 
{ B } )  =  ( { B }  u.  (/) )
5 un0 3612 . . . 4  |-  ( { B }  u.  (/) )  =  { B }
64, 5eqtr2i 2425 . . 3  |-  { B }  =  ( (/)  u.  { B } )
72, 3, 63eqtr4g 2461 . 2  |-  ( { A }  =  (/)  ->  { A ,  B }  =  { B } )
81, 7sylbi 188 1  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278   (/)c0 3588   {csn 3774   {cpr 3775
This theorem is referenced by:  prprc2  3875  prprc  3876  prex  4366  elprchashprn2  11622  usgraedgprv  21349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-un 3285  df-nul 3589  df-sn 3780  df-pr 3781
  Copyright terms: Public domain W3C validator