Theorem sssnm 3717

Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)

Assertion

Ref	Expression
sssnm	|- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem sssnm

Step	Hyp	Ref	Expression
1		ssel 3122	. . . . . . . . . 10 |- ( A C_ { B } -> ( x e. A -> x e. { B } ) )
2		elsni 3578	. . . . . . . . . 10 |- ( x e. { B } -> x = B )
3	1, 2	syl6 33	. . . . . . . . 9 |- ( A C_ { B } -> ( x e. A -> x = B ) )
4		eleq1 2220	. . . . . . . . 9 |- ( x = B -> ( x e. A <-> B e. A ) )
5	3, 4	syl6 33	. . . . . . . 8 |- ( A C_ { B } -> ( x e. A -> ( x e. A <-> B e. A ) ) )
6	5	ibd 177	. . . . . . 7 |- ( A C_ { B } -> ( x e. A -> B e. A ) )
7	6	exlimdv 1799	. . . . . 6 |- ( A C_ { B } -> ( E. x x e. A -> B e. A ) )
8		snssi 3700	. . . . . 6 |- ( B e. A -> { B } C_ A )
9	7, 8	syl6 33	. . . . 5 |- ( A C_ { B } -> ( E. x x e. A -> { B } C_ A ) )
10	9	anc2li 327	. . . 4 |- ( A C_ { B } -> ( E. x x e. A -> ( A C_ { B } /\ { B } C_ A ) ) )
11		eqss 3143	. . . 4 |- ( A = { B } <-> ( A C_ { B } /\ { B } C_ A ) )
12	10, 11	syl6ibr 161	. . 3 |- ( A C_ { B } -> ( E. x x e. A -> A = { B } ) )
13	12	com12 30	. 2 |- ( E. x x e. A -> ( A C_ { B } -> A = { B } ) )
14		eqimss 3182	. 2 |- ( A = { B } -> A C_ { B } )
15	13, 14	impbid1 141	1 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128   C_ wss 3102   {csn 3560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-sn 3566

This theorem is referenced by:  eqsnm  3718  ss1o0el1  4158  exmidn0m  4162  exmidsssn  4163  exmidomni  7085  exmidunben  12166  exmidsbthrlem  13604  sbthomlem  13607