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

Theorem erssxp 7353
Description: An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erssxp

Proof of Theorem erssxp
StepHypRef Expression
1 errel 7339 . . 3
2 relssdmrn 5533 . . 3
31, 2syl 16 . 2
4 erdm 7340 . . 3
5 errn 7352 . . 3
64, 5xpeq12d 5029 . 2
73, 6sseqtrd 3539 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  C_wss 3475  X.cxp 5002  domcdm 5004  rancrn 5005  Relwrel 5009  Erwer 7327
This theorem is referenced by:  erex  7354  riiner  7403  efgval  16735  qtophaus  27839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rn 5015  df-er 7330
  Copyright terms: Public domain W3C validator