Theorem wun0 9540

Description: A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypothesis

Ref	Expression
wun0.1	|- ( ph -> U e. WUni )

Assertion

Ref	Expression
wun0	|- ( ph -> (/) e. U )

Proof of Theorem wun0

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

Step	Hyp	Ref	Expression
1		wun0.1	. . . 4 |- ( ph -> U e. WUni )
2		iswun 9526	. . . . . 6 |- ( 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 ) ) ) )
3	2	ibi 256	. . . . 5 |- ( 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	simp2d 1074	. . . 4 |- ( U e. WUni -> U =/= (/) )
5	1, 4	syl 17	. . 3 |- ( ph -> U =/= (/) )
6		n0 3931	. . 3 |- ( U =/= (/) <-> E. x x e. U )
7	5, 6	sylib 208	. 2 |- ( ph -> E. x x e. U )
8	1	adantr 481	. . 3 |- ( ( ph /\ x e. U ) -> U e. WUni )
9		simpr 477	. . 3 |- ( ( ph /\ x e. U ) -> x e. U )
10		0ss 3972	. . . 4 |- (/) C_ x
11	10	a1i 11	. . 3 |- ( ( ph /\ x e. U ) -> (/) C_ x )
12	8, 9, 11	wunss 9534	. 2 |- ( ( ph /\ x e. U ) -> (/) e. U )
13	7, 12	exlimddv 1863	1 |- ( ph -> (/) e. U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)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  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-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  wunr1om  9541  wunfi  9543  wuntpos  9556  intwun  9557  r1wunlim  9559  wuncval2  9569  wunress  15940  catcoppccl  16758