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Theorem grutr 8658
 Description: A Grothendieck's universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
grutr

Proof of Theorem grutr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elgrug 8657 . . 3
21ibi 233 . 2
32simpld 446 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wcel 1725  wral 2697  cpw 3791  cpr 3807  cuni 4007   wtr 4294   crn 4871  (class class class)co 6073   cmap 7010  cgru 8655 This theorem is referenced by:  gruelss  8659  gruwun  8678  intgru  8679  gruina  8683  grur1  8685  grutsk  8687 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-tr 4295  df-iota 5410  df-fv 5454  df-ov 6076  df-gru 8656
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