Theorem elpwunsn 2908

Description: Membership in an extension of a power class.

Assertion

Ref	Expression
elpwunsn	|- (A e. (P~(B u. {C}) \ P~B) -> C e. A)

Proof of Theorem elpwunsn

Step	Hyp	Ref	Expression
1		eldif 2054	. 2 |- (A e. (P~(B u. {C}) \ P~B) <-> (A e. P~(B u. {C}) /\ -. A e. P~B))
2		elpwg 2402	. . . . . . 7 |- (A e. P~(B u. {C}) -> (A e. P~B <-> A (_ B))
3		dfss3 2056	. . . . . . 7 |- (A (_ B <-> A.x e. A x e. B)
4	2, 3	syl6bb 535	. . . . . 6 |- (A e. P~(B u. {C}) -> (A e. P~B <-> A.x e. A x e. B))
5	4	negbid 610	. . . . 5 |- (A e. P~(B u. {C}) -> (-. A e. P~B <-> -. A.x e. A x e. B))
6	5	biimpa 416	. . . 4 |- ((A e. P~(B u. {C}) /\ -. A e. P~B) -> -. A.x e. A x e. B)
7		rexnal 1652	. . . 4 |- (E.x e. A -. x e. B <-> -. A.x e. A x e. B)
8	6, 7	sylibr 200	. . 3 |- ((A e. P~(B u. {C}) /\ -. A e. P~B) -> E.x e. A -. x e. B)
9		elpwi 2403	. . . . . . . . 9 |- (A e. P~(B u. {C}) -> A (_ (B u. {C}))
10		ssel 2060	. . . . . . . . . 10 |- (A (_ (B u. {C}) -> (x e. A -> x e. (B u. {C})))
11		elun 2170	. . . . . . . . . . . . 13 |- (x e. (B u. {C}) <-> (x e. B \/ x e. {C}))
12		elsni 2429	. . . . . . . . . . . . . . 15 |- (x e. {C} -> x = C)
13	12	orim2i 338	. . . . . . . . . . . . . 14 |- ((x e. B \/ x e. {C}) -> (x e. B \/ x = C))
14	13	ord 232	. . . . . . . . . . . . 13 |- ((x e. B \/ x e. {C}) -> (-. x e. B -> x = C))
15	11, 14	sylbi 199	. . . . . . . . . . . 12 |- (x e. (B u. {C}) -> (-. x e. B -> x = C))
16	15	imim2i 17	. . . . . . . . . . 11 |- ((x e. A -> x e. (B u. {C})) -> (x e. A -> (-. x e. B -> x = C)))
17	16	imp3a 361	. . . . . . . . . 10 |- ((x e. A -> x e. (B u. {C})) -> ((x e. A /\ -. x e. B) -> x = C))
18	10, 17	syl 10	. . . . . . . . 9 |- (A (_ (B u. {C}) -> ((x e. A /\ -. x e. B) -> x = C))
19		eleq1 1532	. . . . . . . . . . 11 |- (x = C -> (x e. A <-> C e. A))
20	19	biimpd 153	. . . . . . . . . 10 |- (x = C -> (x e. A -> C e. A))
21	20	imim2i 17	. . . . . . . . 9 |- (((x e. A /\ -. x e. B) -> x = C) -> ((x e. A /\ -. x e. B) -> (x e. A -> C e. A)))
22	9, 18, 21	3syl 20	. . . . . . . 8 |- (A e. P~(B u. {C}) -> ((x e. A /\ -. x e. B) -> (x e. A -> C e. A)))
23	22	exp3a 375	. . . . . . 7 |- (A e. P~(B u. {C}) -> (x e. A -> (-. x e. B -> (x e. A -> C e. A))))
24	23	com4r 41	. . . . . 6 |- (x e. A -> (A e. P~(B u. {C}) -> (x e. A -> (-. x e. B -> C e. A))))
25	24	pm2.43b 67	. . . . 5 |- (A e. P~(B u. {C}) -> (x e. A -> (-. x e. B -> C e. A)))
26	25	r19.23adv 1744	. . . 4 |- (A e. P~(B u. {C}) -> (E.x e. A -. x e. B -> C e. A))
27	26	imp 350	. . 3 |- ((A e. P~(B u. {C}) /\ E.x e. A -. x e. B) -> C e. A)
28	8, 27	syldan 467	. 2 |- ((A e. P~(B u. {C}) /\ -. A e. P~B) -> C e. A)
29	1, 28	sylbi 199	1 |- (A e. (P~(B u. {C}) \ P~B) -> C e. A)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643  E.wrex 1644   \ cdif 2041   u. cun 2042   (_ wss 2044  P~cpw 2398  {csn 2406

This theorem is referenced by:  pwfilem 4553

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pw 2399  df-sn 2409  df-pr 2410

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