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Theorem gruiin 8675
 Description: A Grothendieck's 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 1629 . . 3
2 nfii1 4114 . . . 4
32nfel1 2581 . . 3
4 iinss2 4135 . . . . . 6
5 gruss 8661 . . . . . 6
64, 5syl3an3 1219 . . . . 5
763exp 1152 . . . 4
87com23 74 . . 3
91, 3, 8rexlimd 2819 . 2
109imp 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725  wrex 2698   wss 3312  ciin 4086  cgru 8655 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  ax-sep 4322 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-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iin 4088  df-br 4205  df-tr 4295  df-iota 5410  df-fv 5454  df-ov 6076  df-gru 8656
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