Theorem wunpr 9531

Description: A weak universe is closed under pairing. (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
wunpr	|- ( ph -> { A , B } e. U )

Proof of Theorem wunpr

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		wunpr.3	. 2 |- ( ph -> B e. U )
3		wununi.1	. . 3 |- ( ph -> U e. WUni )
4		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 ) ) ) )
5	4	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 ) ) )
6	5	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 ) )
7		simp3 1063	. . . 4 |- ( ( U. x e. U /\ ~P x e. U /\ A. y e. U { x , y } e. U ) -> A. y e. U { x , y } e. U )
8	7	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 A. y e. U { x , y } e. U )
9	3, 6, 8	3syl 18	. 2 |- ( ph -> A. x e. U A. y e. U { x , y } e. U )
10		preq1 4268	. . . 4 |- ( x = A -> { x , y } = { A , y } )
11	10	eleq1d 2686	. . 3 |- ( x = A -> ( { x , y } e. U <-> { A , y } e. U ) )
12		preq2 4269	. . . 4 |- ( y = B -> { A , y } = { A , B } )
13	12	eleq1d 2686	. . 3 |- ( y = B -> ( { A , y } e. U <-> { A , B } e. U ) )
14	11, 13	rspc2va 3323	. 2 |- ( ( ( A e. U /\ B e. U ) /\ A. x e. U A. y e. U { x , y } e. U ) -> { A , B } e. U )
15	1, 2, 9, 14	syl21anc 1325	1 |- ( ph -> { A , B } 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-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:  wunun  9532  wuntp  9533  wunsn  9538  wunop  9544  intwun  9557  wuncval2  9569