Theorem wununi 9528

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

Hypotheses

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

Assertion

Ref	Expression
wununi	|- ( ph -> U. A e. U )

Proof of Theorem wununi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wununi.2	. 2 |- ( ph -> A e. U )
2		wununi.1	. . 3 |- ( ph -> U e. WUni )
3		iswun 9526	. . . . 5 |- ( U e. WUni -> ( U e. WUni <-> ( Tr U /\ U =/= (/) /\ A. x e. U ( U. x e. U /\ ~P x e. U /\ A. y e. U { x , y } e. U ) ) ) )
4	3	ibi 256	. . . 4 |- ( U e. WUni -> ( Tr U /\ U =/= (/) /\ A. x e. U ( U. x e. U /\ ~P x e. U /\ A. y e. U { x , y } e. U ) ) )
5	4	simp3d 1075	. . 3 |- ( U e. WUni -> A. x e. U ( U. x e. U /\ ~P x e. U /\ A. y e. U { x , y } e. U ) )
6		simp1 1061	. . . 4 |- ( ( U. x e. U /\ ~P x e. U /\ A. y e. U { x , y } e. U ) -> U. x e. U )
7	6	ralimi 2952	. . 3 |- ( A. x e. U ( U. x e. U /\ ~P x e. U /\ A. y e. U { x , y } e. U ) -> A. x e. U U. x e. U )
8	2, 5, 7	3syl 18	. 2 |- ( ph -> A. x e. U U. x e. U )
9		unieq 4444	. . . 4 |- ( x = A -> U. x = U. A )
10	9	eleq1d 2686	. . 3 |- ( x = A -> ( U. x e. U <-> U. A e. U ) )
11	10	rspcv 3305	. 2 |- ( A e. U -> ( A. x e. U U. x e. U -> U. A e. U ) )
12	1, 8, 11	sylc 65	1 |- ( ph -> U. A e. U )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   ~Pcpw 4158   {cpr 4179   U.cuni 4436   Tr wtr 4752  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-in 3581  df-ss 3588  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  wunun  9532  wunint  9537  wundm  9550  wunrn  9551  wunfv  9554  intwun  9557  wuncval2  9569  wunstr  15881  wunfunc  16559  wunnat  16616  catcoppccl  16758  catcfuccl  16759  catcxpccl  16847