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Theorem gruxp 8684
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 8683 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  e.  U )
2 grupw 8672 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  u.  B )  e.  U )  ->  ~P ( A  u.  B
)  e.  U )
3 grupw 8672 . . . . 5  |-  ( ( U  e.  Univ  /\  ~P ( A  u.  B
)  e.  U )  ->  ~P ~P ( A  u.  B )  e.  U )
4 xpsspw 4988 . . . . . 6  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
5 gruss 8673 . . . . . 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 1269 . . . . 5  |-  ( ( U  e.  Univ  /\  ~P ~P ( A  u.  B
)  e.  U )  ->  ( A  X.  B )  e.  U
)
73, 6syldan 458 . . . 4  |-  ( ( U  e.  Univ  /\  ~P ( A  u.  B
)  e.  U )  ->  ( A  X.  B )  e.  U
)
82, 7syldan 458 . . 3  |-  ( ( U  e.  Univ  /\  ( A  u.  B )  e.  U )  ->  ( A  X.  B )  e.  U )
983ad2antl1 1120 . 2  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  /\  ( A  u.  B
)  e.  U )  ->  ( A  X.  B )  e.  U
)
101, 9mpdan 651 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 937    e. wcel 1726    u. cun 3320    C_ wss 3322   ~Pcpw 3801    X. cxp 4878   Univcgru 8667
This theorem is referenced by:  grumap  8685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-gru 8668
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