Theorem pwsnALT 4429

Description: Alternate proof of pwsn 4428, more direct. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pwsnALT	|- ~P { A } = { (/) , { A } }

Proof of Theorem pwsnALT

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

Step	Hyp	Ref	Expression
1		dfss2 3591	. . . . . . . . 9 |- ( x C_ { A } <-> A. y ( y e. x -> y e. { A } ) )
2		velsn 4193	. . . . . . . . . . 11 |- ( y e. { A } <-> y = A )
3	2	imbi2i 326	. . . . . . . . . 10 |- ( ( y e. x -> y e. { A } ) <-> ( y e. x -> y = A ) )
4	3	albii 1747	. . . . . . . . 9 |- ( A. y ( y e. x -> y e. { A } ) <-> A. y ( y e. x -> y = A ) )
5	1, 4	bitri 264	. . . . . . . 8 |- ( x C_ { A } <-> A. y ( y e. x -> y = A ) )
6		neq0 3930	. . . . . . . . . 10 |- ( -. x = (/) <-> E. y y e. x )
7		exintr 1819	. . . . . . . . . 10 |- ( A. y ( y e. x -> y = A ) -> ( E. y y e. x -> E. y ( y e. x /\ y = A ) ) )
8	6, 7	syl5bi 232	. . . . . . . . 9 |- ( A. y ( y e. x -> y = A ) -> ( -. x = (/) -> E. y ( y e. x /\ y = A ) ) )
9		df-clel 2618	. . . . . . . . . . 11 |- ( A e. x <-> E. y ( y = A /\ y e. x ) )
10		exancom 1787	. . . . . . . . . . 11 |- ( E. y ( y = A /\ y e. x ) <-> E. y ( y e. x /\ y = A ) )
11	9, 10	bitr2i 265	. . . . . . . . . 10 |- ( E. y ( y e. x /\ y = A ) <-> A e. x )
12		snssi 4339	. . . . . . . . . 10 |- ( A e. x -> { A } C_ x )
13	11, 12	sylbi 207	. . . . . . . . 9 |- ( E. y ( y e. x /\ y = A ) -> { A } C_ x )
14	8, 13	syl6 35	. . . . . . . 8 |- ( A. y ( y e. x -> y = A ) -> ( -. x = (/) -> { A } C_ x ) )
15	5, 14	sylbi 207	. . . . . . 7 |- ( x C_ { A } -> ( -. x = (/) -> { A } C_ x ) )
16	15	anc2li 580	. . . . . 6 |- ( x C_ { A } -> ( -. x = (/) -> ( x C_ { A } /\ { A } C_ x ) ) )
17		eqss 3618	. . . . . 6 |- ( x = { A } <-> ( x C_ { A } /\ { A } C_ x ) )
18	16, 17	syl6ibr 242	. . . . 5 |- ( x C_ { A } -> ( -. x = (/) -> x = { A } ) )
19	18	orrd 393	. . . 4 |- ( x C_ { A } -> ( x = (/) \/ x = { A } ) )
20		0ss 3972	. . . . . 6 |- (/) C_ { A }
21		sseq1 3626	. . . . . 6 |- ( x = (/) -> ( x C_ { A } <-> (/) C_ { A } ) )
22	20, 21	mpbiri 248	. . . . 5 |- ( x = (/) -> x C_ { A } )
23		eqimss 3657	. . . . 5 |- ( x = { A } -> x C_ { A } )
24	22, 23	jaoi 394	. . . 4 |- ( ( x = (/) \/ x = { A } ) -> x C_ { A } )
25	19, 24	impbii 199	. . 3 |- ( x C_ { A } <-> ( x = (/) \/ x = { A } ) )
26	25	abbii 2739	. 2 |- { x | x C_ { A } } = { x | ( x = (/) \/ x = { A } ) }
27		df-pw 4160	. 2 |- ~P { A } = { x | x C_ { A } }
28		dfpr2 4195	. 2 |- { (/) , { A } } = { x | ( x = (/) \/ x = { A } ) }
29	26, 27, 28	3eqtr4i 2654	1 |- ~P { A } = { (/) , { A } }

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179

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-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)