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Theorem wunxp 9005
Description: A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1  |-  ( ph  ->  U  e. WUni )
wunop.2  |-  ( ph  ->  A  e.  U )
wunop.3  |-  ( ph  ->  B  e.  U )
Assertion
Ref Expression
wunxp  |-  ( ph  ->  ( A  X.  B
)  e.  U )

Proof of Theorem wunxp
StepHypRef Expression
1 wun0.1 . 2  |-  ( ph  ->  U  e. WUni )
2 wunop.2 . . . . 5  |-  ( ph  ->  A  e.  U )
3 wunop.3 . . . . 5  |-  ( ph  ->  B  e.  U )
41, 2, 3wunun 8991 . . . 4  |-  ( ph  ->  ( A  u.  B
)  e.  U )
51, 4wunpw 8988 . . 3  |-  ( ph  ->  ~P ( A  u.  B )  e.  U
)
61, 5wunpw 8988 . 2  |-  ( ph  ->  ~P ~P ( A  u.  B )  e.  U )
7 xpsspw 5064 . . 3  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
87a1i 11 . 2  |-  ( ph  ->  ( A  X.  B
)  C_  ~P ~P ( A  u.  B
) )
91, 6, 8wunss 8993 1  |-  ( ph  ->  ( A  X.  B
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758    u. cun 3437    C_ wss 3439   ~Pcpw 3971    X. cxp 4949  WUnicwun 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-opab 4462  df-tr 4497  df-xp 4957  df-wun 8983
This theorem is referenced by:  wunpm  9006  wuncnv  9011  wunco  9014  wuntpos  9015  tskxp  9068  wuncn  9451  wunfunc  14931  wunnat  14988  catcoppccl  15098  catcfuccl  15099  catcxpccl  15139
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