Theorem wunun 9532

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

Hypotheses

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

Assertion

Ref	Expression
wunun	|- ( ph -> ( A u. B ) e. U )

Proof of Theorem wunun

Step	Hyp	Ref	Expression
1		wununi.2	. . 3 |- ( ph -> A e. U )
2		wunpr.3	. . 3 |- ( ph -> B e. U )
3		uniprg 4450	. . 3 |- ( ( A e. U /\ B e. U ) -> U. { A , B } = ( A u. B ) )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> U. { A , B } = ( A u. B ) )
5		wununi.1	. . 3 |- ( ph -> U e. WUni )
6	5, 1, 2	wunpr 9531	. . 3 |- ( ph -> { A , B } e. U )
7	5, 6	wununi 9528	. 2 |- ( ph -> U. { A , B } e. U )
8	4, 7	eqeltrrd 2702	1 |- ( ph -> ( A u. B ) e. U )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   {cpr 4179   U.cuni 4436  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

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-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  wuntp  9533  wunsuc  9539  wunfi  9543  wunxp  9546  wuntpos  9556  wunsets  15900  catcoppccl  16758