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Theorem wunxp 9091
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 9077 . . . 4  |-  ( ph  ->  ( A  u.  B
)  e.  U )
51, 4wunpw 9074 . . 3  |-  ( ph  ->  ~P ( A  u.  B )  e.  U
)
61, 5wunpw 9074 . 2  |-  ( ph  ->  ~P ~P ( A  u.  B )  e.  U )
7 xpsspw 5104 . . 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 9079 1  |-  ( ph  ->  ( A  X.  B
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823    u. cun 3459    C_ wss 3461   ~Pcpw 3999    X. cxp 4986  WUnicwun 9067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-opab 4498  df-tr 4533  df-xp 4994  df-wun 9069
This theorem is referenced by:  wunpm  9092  wuncnv  9097  wunco  9100  wuntpos  9101  tskxp  9154  wuncn  9536  wunfunc  15390  wunnat  15447  catcoppccl  15589  catcfuccl  15590  catcxpccl  15678
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