Theorem iswun 9526

Description: Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
iswun	|- ( U e. V -> ( 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 ) ) ) )

Distinct variable group:   x, y, U

Allowed substitution hints:   V( x, y)

Proof of Theorem iswun

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		treq 4758	. . 3 |- ( u = U -> ( Tr u <-> Tr U ) )
2		neeq1 2856	. . 3 |- ( u = U -> ( u =/= (/) <-> U =/= (/) ) )
3		eleq2 2690	. . . . 5 |- ( u = U -> ( U. x e. u <-> U. x e. U ) )
4		eleq2 2690	. . . . 5 |- ( u = U -> ( ~P x e. u <-> ~P x e. U ) )
5		eleq2 2690	. . . . . 6 |- ( u = U -> ( { x , y } e. u <-> { x , y } e. U ) )
6	5	raleqbi1dv 3146	. . . . 5 |- ( u = U -> ( A. y e. u { x , y } e. u <-> A. y e. U { x , y } e. U ) )
7	3, 4, 6	3anbi123d 1399	. . . 4 |- ( u = U -> ( ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u ) <-> ( U. x e. U /\ ~P x e. U /\ A. y e. U { x , y } e. U ) ) )
8	7	raleqbi1dv 3146	. . 3 |- ( u = U -> ( 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 /\ ~P x e. U /\ A. y e. U { x , y } e. U ) ) )
9	1, 2, 8	3anbi123d 1399	. 2 |- ( u = U -> ( ( Tr u /\ u =/= (/) /\ A. x e. u ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u ) ) <-> ( Tr U /\ U =/= (/) /\ A. x e. U ( U. x e. U /\ ~P x e. U /\ A. y e. U { x , y } e. U ) ) ) )
10		df-wun 9524	. 2 |- WUni = { u | ( Tr u /\ u =/= (/) /\ A. x e. u ( U. x e. u /\ ~P x e. u /\ A. y e. u { x , y } e. u ) ) }
11	9, 10	elab2g 3353	1 |- ( U e. V -> ( 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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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:  wuntr  9527  wununi  9528  wunpw  9529  wunpr  9531  wun0  9540  intwun  9557  r1limwun  9558  wunex2  9560  tskwun  9606  gruwun  9635  pwinfi2  37867