Theorem wunxp 9546

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

Step	Hyp	Ref	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 )
4	1, 2, 3	wunun 9532	. . . 4 |- ( ph -> ( A u. B ) e. U )
5	1, 4	wunpw 9529	. . 3 |- ( ph -> ~P ( A u. B ) e. U )
6	1, 5	wunpw 9529	. 2 |- ( ph -> ~P ~P ( A u. B ) e. U )
7		xpsspw 5233	. . 3 |- ( A X. B ) C_ ~P ~P ( A u. B )
8	7	a1i 11	. 2 |- ( ph -> ( A X. B ) C_ ~P ~P ( A u. B ) )
9	1, 6, 8	wunss 9534	1 |- ( ph -> ( A X. B ) e. U )

Syntax hints:   -> wi 4   e. wcel 1990   u. cun 3572   C_ wss 3574   ~Pcpw 4158   X. cxp 5112  WUnicwun 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-opab 4713  df-tr 4753  df-xp 5120  df-rel 5121  df-wun 9524

This theorem is referenced by:  wunpm  9547  wuncnv  9552  wunco  9555  wuntpos  9556  tskxp  9609  wuncn  9991  wunfunc  16559  wunnat  16616  catcoppccl  16758  catcfuccl  16759  catcxpccl  16847