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Theorem gruxp 8429
Description: A Grothendieck's universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruxp  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  X.  B )  e.  U )

Proof of Theorem gruxp
StepHypRef Expression
1 gruun 8428 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  e.  U )
2 grupw 8417 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  u.  B )  e.  U )  ->  ~P ( A  u.  B
)  e.  U )
3 grupw 8417 . . . . 5  |-  ( ( U  e.  Univ  /\  ~P ( A  u.  B
)  e.  U )  ->  ~P ~P ( A  u.  B )  e.  U )
4 xpsspw 4797 . . . . . 6  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
5 gruss 8418 . . . . . 6  |-  ( ( U  e.  Univ  /\  ~P ~P ( A  u.  B
)  e.  U  /\  ( A  X.  B
)  C_  ~P ~P ( A  u.  B
) )  ->  ( A  X.  B )  e.  U )
64, 5mp3an3 1266 . . . . 5  |-  ( ( U  e.  Univ  /\  ~P ~P ( A  u.  B
)  e.  U )  ->  ( A  X.  B )  e.  U
)
73, 6syldan 456 . . . 4  |-  ( ( U  e.  Univ  /\  ~P ( A  u.  B
)  e.  U )  ->  ( A  X.  B )  e.  U
)
82, 7syldan 456 . . 3  |-  ( ( U  e.  Univ  /\  ( A  u.  B )  e.  U )  ->  ( A  X.  B )  e.  U )
983ad2antl1 1117 . 2  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  /\  ( A  u.  B
)  e.  U )  ->  ( A  X.  B )  e.  U
)
101, 9mpdan 649 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  X.  B )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684    u. cun 3150    C_ wss 3152   ~Pcpw 3625    X. cxp 4687   Univcgru 8412
This theorem is referenced by:  grumap  8430
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-gru 8413
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