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

Theorem gruelss 9193
Description: A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruelss

Proof of Theorem gruelss
StepHypRef Expression
1 grutr 9192 . 2
2 trss 4554 . . 3
32imp 429 . 2
41, 3sylan 471 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  C_wss 3475  Trwtr 4545   cgru 9189
This theorem is referenced by:  gruss  9195  gruuni  9199  gruel  9202  grur1a  9218  grur1  9219
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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  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-uni 4250  df-br 4453  df-tr 4546  df-iota 5556  df-fv 5601  df-ov 6299  df-gru 9190
  Copyright terms: Public domain W3C validator