Theorem sssn 4358

Description: The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
sssn	|- ( A C_ { B } <-> ( A = (/) \/ A = { B } ) )

Proof of Theorem sssn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 3930	. . . . . . 7 |- ( -. A = (/) <-> E. x x e. A )
2		ssel 3597	. . . . . . . . . . 11 |- ( A C_ { B } -> ( x e. A -> x e. { B } ) )
3		elsni 4194	. . . . . . . . . . 11 |- ( x e. { B } -> x = B )
4	2, 3	syl6 35	. . . . . . . . . 10 |- ( A C_ { B } -> ( x e. A -> x = B ) )
5		eleq1 2689	. . . . . . . . . 10 |- ( x = B -> ( x e. A <-> B e. A ) )
6	4, 5	syl6 35	. . . . . . . . 9 |- ( A C_ { B } -> ( x e. A -> ( x e. A <-> B e. A ) ) )
7	6	ibd 258	. . . . . . . 8 |- ( A C_ { B } -> ( x e. A -> B e. A ) )
8	7	exlimdv 1861	. . . . . . 7 |- ( A C_ { B } -> ( E. x x e. A -> B e. A ) )
9	1, 8	syl5bi 232	. . . . . 6 |- ( A C_ { B } -> ( -. A = (/) -> B e. A ) )
10		snssi 4339	. . . . . 6 |- ( B e. A -> { B } C_ A )
11	9, 10	syl6 35	. . . . 5 |- ( A C_ { B } -> ( -. A = (/) -> { B } C_ A ) )
12	11	anc2li 580	. . . 4 |- ( A C_ { B } -> ( -. A = (/) -> ( A C_ { B } /\ { B } C_ A ) ) )
13		eqss 3618	. . . 4 |- ( A = { B } <-> ( A C_ { B } /\ { B } C_ A ) )
14	12, 13	syl6ibr 242	. . 3 |- ( A C_ { B } -> ( -. A = (/) -> A = { B } ) )
15	14	orrd 393	. 2 |- ( A C_ { B } -> ( A = (/) \/ A = { B } ) )
16		0ss 3972	. . . 4 |- (/) C_ { B }
17		sseq1 3626	. . . 4 |- ( A = (/) -> ( A C_ { B } <-> (/) C_ { B } ) )
18	16, 17	mpbiri 248	. . 3 |- ( A = (/) -> A C_ { B } )
19		eqimss 3657	. . 3 |- ( A = { B } -> A C_ { B } )
20	18, 19	jaoi 394	. 2 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )
21	15, 20	impbii 199	1 |- ( A C_ { B } <-> ( A = (/) \/ A = { B } ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   C_ wss 3574   (/)c0 3915   {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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178

This theorem is referenced by:  eqsn  4361  eqsnOLD  4362  snsssn  4372  pwsn  4428  frsn  5189  foconst  6126  fin1a2lem12  9233  fpwwe2lem13  9464  gsumval2  17280  0top  20787  minveclem4a  23201  uvtxa01vtx0  26297  locfinref  29908  ordcmp  32446  bj-snmoore  33068  uneqsn  38321