Theorem sssnm 3784

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 3177	. . . . . . . . . 10 |- ( A C_ { B } -> ( x e. A -> x e. { B } ) )
2		elsni 3640	. . . . . . . . . 10 |- ( x e. { B } -> x = B )
3	1, 2	syl6 33	. . . . . . . . 9 |- ( A C_ { B } -> ( x e. A -> x = B ) )
4		eleq1 2259	. . . . . . . . 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 178	. . . . . . 7 |- ( A C_ { B } -> ( x e. A -> B e. A ) )
7	6	exlimdv 1833	. . . . . 6 |- ( A C_ { B } -> ( E. x x e. A -> B e. A ) )
8		snssi 3766	. . . . . 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 329	. . . 4 |- ( A C_ { B } -> ( E. x x e. A -> ( A C_ { B } /\ { B } C_ A ) ) )
11		eqss 3198	. . . 4 |- ( A = { B } <-> ( A C_ { B } /\ { B } C_ A ) )
12	10, 11	imbitrrdi 162	. . 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 3237	. 2 |- ( A = { B } -> A C_ { B } )
15	13, 14	impbid1 142	1 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   C_ wss 3157   {csn 3622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-sn 3628

This theorem is referenced by:  eqsnm  3785  ss1o0el1  4230  exmidn0m  4234  exmidsssn  4235  exmidomni  7208  exmidunben  12643  exmidsbthrlem  15666  sbthomlem  15669