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

Theorem gruelss 9237
 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 9236 . 2
2 trss 4499 . . 3
32imp 436 . 2
41, 3sylan 479 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wcel 1904   wss 3390   wtr 4490  cgru 9233 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-tr 4491  df-iota 5553  df-fv 5597  df-ov 6311  df-gru 9234 This theorem is referenced by:  gruss  9239  gruuni  9243  gruel  9246  grur1a  9262  grur1  9263
 Copyright terms: Public domain W3C validator