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Theorem grupr 9187
 Description: A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grupr

Proof of Theorem grupr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elgrug 9182 . . . . . . 7
21ibi 241 . . . . . 6
32simprd 463 . . . . 5
4 preq2 4113 . . . . . . . . . 10
54eleq1d 2536 . . . . . . . . 9
65rspccv 3216 . . . . . . . 8
763ad2ant2 1018 . . . . . . 7
87com12 31 . . . . . 6
98ralimdv 2877 . . . . 5
103, 9syl5com 30 . . . 4
11 preq1 4112 . . . . . 6
1211eleq1d 2536 . . . . 5
1312rspccv 3216 . . . 4
1410, 13syl6 33 . . 3
1514com23 78 . 2
16153imp 1190 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379   wcel 1767  wral 2817  cpw 4016  cpr 4035  cuni 4251   wtr 4546   crn 5006  (class class class)co 6295   cmap 7432  cgru 9180 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-tr 4547  df-iota 5557  df-fv 5602  df-ov 6298  df-gru 9181 This theorem is referenced by:  grusn  9194  gruop  9195  gruun  9196  gruwun  9203  intgru  9204
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