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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  |-  ( ( U  e.  Univ  /\  E. x  e.  A  B  e.  U )  ->  |^|_ x  e.  A  B  e.  U )
Distinct variable groups:    x, U    x, A
Allowed substitution hint:    B( x)

Proof of Theorem gruiin
StepHypRef Expression
1 nfv 1629 . . 3  |-  F/ x  U  e.  Univ
2 nfii1 4114 . . . 4  |-  F/_ x |^|_ x  e.  A  B
32nfel1 2581 . . 3  |-  F/ x |^|_ x  e.  A  B  e.  U
4 iinss2 4135 . . . . . 6  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 gruss 8661 . . . . . 6  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  |^|_ x  e.  A  B  C_  B
)  ->  |^|_ x  e.  A  B  e.  U
)
64, 5syl3an3 1219 . . . . 5  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  x  e.  A )  ->  |^|_ x  e.  A  B  e.  U )
763exp 1152 . . . 4  |-  ( U  e.  Univ  ->  ( B  e.  U  ->  (
x  e.  A  ->  |^|_ x  e.  A  B  e.  U ) ) )
87com23 74 . . 3  |-  ( U  e.  Univ  ->  ( x  e.  A  ->  ( B  e.  U  ->  |^|_
x  e.  A  B  e.  U ) ) )
91, 3, 8rexlimd 2819 . 2  |-  ( U  e.  Univ  ->  ( E. x  e.  A  B  e.  U  ->  |^|_ x  e.  A  B  e.  U ) )
109imp 419 1  |-  ( ( U  e.  Univ  /\  E. x  e.  A  B  e.  U )  ->  |^|_ x  e.  A  B  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   E.wrex 2698    C_ wss 3312   |^|_ciin 4086   Univcgru 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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