Theorem wunss 9534

Description: A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	|- ( ph -> U e. WUni )
wununi.2	|- ( ph -> A e. U )
wunss.3	|- ( ph -> B C_ A )

Assertion

Ref	Expression
wunss	|- ( ph -> B e. U )

Proof of Theorem wunss

Step	Hyp	Ref	Expression
1		wununi.1	. . 3 |- ( ph -> U e. WUni )
2		wununi.2	. . . 4 |- ( ph -> A e. U )
3	1, 2	wunpw 9529	. . 3 |- ( ph -> ~P A e. U )
4	1, 3	wunelss 9530	. 2 |- ( ph -> ~P A C_ U )
5		wunss.3	. . 3 |- ( ph -> B C_ A )
6	2, 5	sselpwd 4807	. 2 |- ( ph -> B e. ~P A )
7	4, 6	sseldd 3604	1 |- ( ph -> B e. U )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   ~Pcpw 4158  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

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-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  wunin  9535  wundif  9536  wunint  9537  wun0  9540  wunom  9542  wunxp  9546  wunpm  9547  wunmap  9548  wundm  9550  wunrn  9551  wuncnv  9552  wunres  9553  wunfv  9554  wunco  9555  wuntpos  9556  wuncn  9991  wunndx  15878  wunstr  15881  wunfunc  16559