Theorem elpwdifsn 4319

Description: A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.)

Assertion

Ref	Expression
elpwdifsn	|- ( ( S e. W /\ S C_ V /\ A e/ S ) -> S e. ~P ( V \ { A } ) )

Proof of Theorem elpwdifsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . . . 6 |- ( ( S e. W /\ S C_ V /\ A e/ S ) -> S C_ V )
2	1	sselda 3603	. . . . 5 |- ( ( ( S e. W /\ S C_ V /\ A e/ S ) /\ x e. S ) -> x e. V )
3		df-nel 2898	. . . . . . . . . 10 |- ( A e/ S <-> -. A e. S )
4	3	biimpi 206	. . . . . . . . 9 |- ( A e/ S -> -. A e. S )
5	4	3ad2ant3 1084	. . . . . . . 8 |- ( ( S e. W /\ S C_ V /\ A e/ S ) -> -. A e. S )
6	5	anim1i 592	. . . . . . 7 |- ( ( ( S e. W /\ S C_ V /\ A e/ S ) /\ x e. S ) -> ( -. A e. S /\ x e. S ) )
7	6	ancomd 467	. . . . . 6 |- ( ( ( S e. W /\ S C_ V /\ A e/ S ) /\ x e. S ) -> ( x e. S /\ -. A e. S ) )
8		nelne2 2891	. . . . . 6 |- ( ( x e. S /\ -. A e. S ) -> x =/= A )
9	7, 8	syl 17	. . . . 5 |- ( ( ( S e. W /\ S C_ V /\ A e/ S ) /\ x e. S ) -> x =/= A )
10		eldifsn 4317	. . . . 5 |- ( x e. ( V \ { A } ) <-> ( x e. V /\ x =/= A ) )
11	2, 9, 10	sylanbrc 698	. . . 4 |- ( ( ( S e. W /\ S C_ V /\ A e/ S ) /\ x e. S ) -> x e. ( V \ { A } ) )
12	11	ex 450	. . 3 |- ( ( S e. W /\ S C_ V /\ A e/ S ) -> ( x e. S -> x e. ( V \ { A } ) ) )
13	12	ssrdv 3609	. 2 |- ( ( S e. W /\ S C_ V /\ A e/ S ) -> S C_ ( V \ { A } ) )
14		elpwg 4166	. . 3 |- ( S e. W -> ( S e. ~P ( V \ { A } ) <-> S C_ ( V \ { A } ) ) )
15	14	3ad2ant1 1082	. 2 |- ( ( S e. W /\ S C_ V /\ A e/ S ) -> ( S e. ~P ( V \ { A } ) <-> S C_ ( V \ { A } ) ) )
16	13, 15	mpbird 247	1 |- ( ( S e. W /\ S C_ V /\ A e/ S ) -> S e. ~P ( V \ { A } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   e/ wnel 2897   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   {csn 4177

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-nel 2898  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pw 4160  df-sn 4178

This theorem is referenced by:  uhgrspan1  26195  upgrreslem  26196  umgrreslem  26197  umgrres1lem  26202  upgrres1  26205