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

Theorem prprc1 3916
 Description: A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
prprc1

Proof of Theorem prprc1
StepHypRef Expression
1 snprc 3873 . 2
2 uneq1 3496 . . 3
3 df-pr 3823 . . 3
4 uncom 3493 . . . 4
5 un0 3654 . . . 4
64, 5eqtr2i 2459 . . 3
72, 3, 63eqtr4g 2495 . 2
81, 7sylbi 189 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wceq 1653   wcel 1726  cvv 2958   cun 3320  c0 3630  csn 3816  cpr 3817 This theorem is referenced by:  prprc2  3917  prprc  3918  prex  4409  elprchashprn2  11672  usgraedgprv  21401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-nul 3631  df-sn 3822  df-pr 3823
 Copyright terms: Public domain W3C validator