Theorem sssnm 3525

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 2939	. . . . . . . . . 10 |- ( A C_ { B } -> ( x e. A -> x e. { B } ) )
2		elsni 3393	. . . . . . . . . 10 |- ( x e. { B } -> x = B )
3	1, 2	syl6 29	. . . . . . . . 9 |- ( A C_ { B } -> ( x e. A -> x = B ) )
4		eleq1 2100	. . . . . . . . 9 |- ( x = B -> ( x e. A <-> B e. A ) )
5	3, 4	syl6 29	. . . . . . . 8 |- ( A C_ { B } -> ( x e. A -> ( x e. A <-> B e. A ) ) )
6	5	ibd 167	. . . . . . 7 |- ( A C_ { B } -> ( x e. A -> B e. A ) )
7	6	exlimdv 1700	. . . . . 6 |- ( A C_ { B } -> ( E. x x e. A -> B e. A ) )
8		snssi 3508	. . . . . 6 |- ( B e. A -> { B } C_ A )
9	7, 8	syl6 29	. . . . 5 |- ( A C_ { B } -> ( E. x x e. A -> { B } C_ A ) )
10	9	anc2li 312	. . . 4 |- ( A C_ { B } -> ( E. x x e. A -> ( A C_ { B } /\ { B } C_ A ) ) )
11		eqss 2960	. . . 4 |- ( A = { B } <-> ( A C_ { B } /\ { B } C_ A ) )
12	10, 11	syl6ibr 151	. . 3 |- ( A C_ { B } -> ( E. x x e. A -> A = { B } ) )
13	12	com12 27	. 2 |- ( E. x x e. A -> ( A C_ { B } -> A = { B } ) )
14		eqimss 2997	. 2 |- ( A = { B } -> A C_ { B } )
15	13, 14	impbid1 130	1 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-sn 3381

This theorem is referenced by:  eqsnm  3526