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Theorem xpider 7401
 Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
xpider

Proof of Theorem xpider
StepHypRef Expression
1 relxp 5115 . 2
2 dmxpid 5227 . 2
3 cnvxp 5429 . . 3
4 xpidtr 5394 . . 3
5 uneq1 3650 . . . 4
6 unss2 3674 . . . 4
7 unidm 3646 . . . . 5
8 eqtr 2483 . . . . . 6
9 sseq2 3525 . . . . . . 7
109biimpd 207 . . . . . 6
118, 10syl 16 . . . . 5
127, 11mpan2 671 . . . 4
135, 6, 12syl2im 38 . . 3
143, 4, 13mp2 9 . 2
15 df-er 7330 . 2
161, 2, 14, 15mpbir3an 1178 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  u.cun 3473  C_wss 3475  X.cxp 5002  'ccnv 5003  domcdm 5004  o.ccom 5008  Relwrel 5009  Er`wer 7327 This theorem is referenced by:  riiner  7403  efglem  16734 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-co 5013  df-dm 5014  df-er 7330
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