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Theorem gruiin 8448
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 1609 . . 3  |-  F/ x  U  e.  Univ
2 nfii1 3950 . . . 4  |-  F/_ x |^|_ x  e.  A  B
32nfel1 2442 . . 3  |-  F/ x |^|_ x  e.  A  B  e.  U
4 iinss2 3970 . . . . . 6  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 gruss 8434 . . . . . 6  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  |^|_ x  e.  A  B  C_  B
)  ->  |^|_ x  e.  A  B  e.  U
)
64, 5syl3an3 1217 . . . . 5  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  x  e.  A )  ->  |^|_ x  e.  A  B  e.  U )
763exp 1150 . . . 4  |-  ( U  e.  Univ  ->  ( B  e.  U  ->  (
x  e.  A  ->  |^|_ x  e.  A  B  e.  U ) ) )
87com23 72 . . 3  |-  ( U  e.  Univ  ->  ( x  e.  A  ->  ( B  e.  U  ->  |^|_
x  e.  A  B  e.  U ) ) )
91, 3, 8rexlimd 2677 . 2  |-  ( U  e.  Univ  ->  ( E. x  e.  A  B  e.  U  ->  |^|_ x  e.  A  B  e.  U ) )
109imp 418 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 358    e. wcel 1696   E.wrex 2557    C_ wss 3165   |^|_ciin 3922   Univcgru 8428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iin 3924  df-br 4040  df-tr 4130  df-iota 5235  df-fv 5279  df-ov 5877  df-gru 8429
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