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Theorem gruiin 8620
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 1626 . . 3  |-  F/ x  U  e.  Univ
2 nfii1 4066 . . . 4  |-  F/_ x |^|_ x  e.  A  B
32nfel1 2535 . . 3  |-  F/ x |^|_ x  e.  A  B  e.  U
4 iinss2 4086 . . . . . 6  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 gruss 8606 . . . . . 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 2772 . 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 1717   E.wrex 2652    C_ wss 3265   |^|_ciin 4038   Univcgru 8600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iin 4040  df-br 4156  df-tr 4246  df-iota 5360  df-fv 5404  df-ov 6025  df-gru 8601
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