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Theorem elgrug 8659
 Description: Properties of a Grothendieck's universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
elgrug
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem elgrug
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 treq 4300 . . 3
2 eleq2 2496 . . . . 5
3 eleq2 2496 . . . . . 6
43raleqbi1dv 2904 . . . . 5
5 oveq1 6080 . . . . . 6
6 eleq2 2496 . . . . . 6
75, 6raleqbidv 2908 . . . . 5
82, 4, 73anbi123d 1254 . . . 4
98raleqbi1dv 2904 . . 3
101, 9anbi12d 692 . 2
11 df-gru 8658 . 2
1210, 11elab2g 3076 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  cpw 3791  cpr 3807  cuni 4007   wtr 4294   crn 4871  (class class class)co 6073   cmap 7010  cgru 8657 This theorem is referenced by:  grutr  8660  grupw  8662  grupr  8664  gruurn  8665  intgru  8681  ingru  8682  grutsk1  8688 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 8658
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