Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ertr4d Structured version   Unicode version

Theorem ertr4d 6927
 Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ersymb.1
ertr4d.5
ertr4d.6
Assertion
Ref Expression
ertr4d

Proof of Theorem ertr4d
StepHypRef Expression
1 ersymb.1 . 2
2 ertr4d.5 . 2
3 ertr4d.6 . . 3
41, 3ersym 6920 . 2
51, 2, 4ertrd 6924 1
 Colors of variables: wff set class Syntax hints:   wi 4   class class class wbr 4215   wer 6905 This theorem is referenced by:  erref  6928  erdisj  6955  nqereu  8811  nqereq  8817  efgredeu  15389  pi1xfr  19085  pi1xfrcnvlem  19086 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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-er 6908
 Copyright terms: Public domain W3C validator