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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  |-  ( ( 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 1683 . . 3  |-  F/ x  U  e.  Univ
2 nfii1 4356 . . . 4  |-  F/_ x |^|_ x  e.  A  B
32nfel1 2645 . . 3  |-  F/ x |^|_ x  e.  A  B  e.  U
4 iinss2 4377 . . . . . 6  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 gruss 9170 . . . . . 6  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  |^|_ x  e.  A  B  C_  B
)  ->  |^|_ x  e.  A  B  e.  U
)
64, 5syl3an3 1263 . . . . 5  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  x  e.  A )  ->  |^|_ x  e.  A  B  e.  U )
763exp 1195 . . . 4  |-  ( U  e.  Univ  ->  ( B  e.  U  ->  (
x  e.  A  ->  |^|_ x  e.  A  B  e.  U ) ) )
87com23 78 . . 3  |-  ( U  e.  Univ  ->  ( x  e.  A  ->  ( B  e.  U  ->  |^|_
x  e.  A  B  e.  U ) ) )
91, 3, 8rexlimd 2947 . 2  |-  ( U  e.  Univ  ->  ( E. x  e.  A  B  e.  U  ->  |^|_ x  e.  A  B  e.  U ) )
109imp 429 1  |-  ( ( U  e.  Univ  /\  E. x  e.  A  B  e.  U )  ->  |^|_ x  e.  A  B  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   E.wrex 2815    C_ wss 3476   |^|_ciin 4326   Univcgru 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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