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Theorem gruiin 9184
 Description: A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruiin
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem gruiin
StepHypRef Expression
1 nfv 1683 . . 3
2 nfii1 4356 . . . 4
32nfel1 2645 . . 3
4 iinss2 4377 . . . . . 6
5 gruss 9170 . . . . . 6
64, 5syl3an3 1263 . . . . 5
763exp 1195 . . . 4
87com23 78 . . 3
91, 3, 8rexlimd 2947 . 2
109imp 429 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wcel 1767  wrex 2815   wss 3476  ciin 4326  cgru 9164 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  ax-sep 4568 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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iin 4328  df-br 4448  df-tr 4541  df-iota 5549  df-fv 5594  df-ov 6285  df-gru 9165 This theorem is referenced by: (None)
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