MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gruiin Unicode version

Theorem gruiin 8432
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 1605 . . 3  |-  F/ x  U  e.  Univ
2 nfii1 3934 . . . 4  |-  F/_ x |^|_ x  e.  A  B
32nfel1 2429 . . 3  |-  F/ x |^|_ x  e.  A  B  e.  U
4 iinss2 3954 . . . . . 6  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 gruss 8418 . . . . . 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 2664 . 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 1684   E.wrex 2544    C_ wss 3152   |^|_ciin 3906   Univcgru 8412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iin 3908  df-br 4024  df-tr 4114  df-iota 5219  df-fv 5263  df-ov 5861  df-gru 8413
  Copyright terms: Public domain W3C validator