Theorem grupw 9617

Description: A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
grupw	|- ( ( U e. Univ /\ A e. U ) -> ~P A e. U )

Proof of Theorem grupw

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

Step	Hyp	Ref	Expression
1		elgrug 9614	. . . . 5 |- ( U e. Univ -> ( U e. Univ <-> ( Tr U /\ A. y e. U ( ~P y e. U /\ A. x e. U { y , x } e. U /\ A. x e. ( U ^m y ) U. ran x e. U ) ) ) )
2	1	ibi 256	. . . 4 |- ( U e. Univ -> ( Tr U /\ A. y e. U ( ~P y e. U /\ A. x e. U { y , x } e. U /\ A. x e. ( U ^m y ) U. ran x e. U ) ) )
3	2	simprd 479	. . 3 |- ( U e. Univ -> A. y e. U ( ~P y e. U /\ A. x e. U { y , x } e. U /\ A. x e. ( U ^m y ) U. ran x e. U ) )
4		simp1 1061	. . . 4 |- ( ( ~P y e. U /\ A. x e. U { y , x } e. U /\ A. x e. ( U ^m y ) U. ran x e. U ) -> ~P y e. U )
5	4	ralimi 2952	. . 3 |- ( A. y e. U ( ~P y e. U /\ A. x e. U { y , x } e. U /\ A. x e. ( U ^m y ) U. ran x e. U ) -> A. y e. U ~P y e. U )
6		pweq 4161	. . . . 5 |- ( y = A -> ~P y = ~P A )
7	6	eleq1d 2686	. . . 4 |- ( y = A -> ( ~P y e. U <-> ~P A e. U ) )
8	7	rspccv 3306	. . 3 |- ( A. y e. U ~P y e. U -> ( A e. U -> ~P A e. U ) )
9	3, 5, 8	3syl 18	. 2 |- ( U e. Univ -> ( A e. U -> ~P A e. U ) )
10	9	imp 445	1 |- ( ( U e. Univ /\ A e. U ) -> ~P A e. U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ~Pcpw 4158   {cpr 4179   U.cuni 4436   Tr wtr 4752   ran crn 5115  (class class class)co 6650   ^m cmap 7857   Univcgru 9612

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-tr 4753  df-iota 5851  df-fv 5896  df-ov 6653  df-gru 9613

This theorem is referenced by:  gruss  9618  grurn  9623  gruxp  9629  grumap  9630  gruwun  9635  intgru  9636  gruina  9640  grur1a  9641