Theorem sssnm 3756

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 3151	. . . . . . . . . 10 |- ( A C_ { B } -> ( x e. A -> x e. { B } ) )
2		elsni 3612	. . . . . . . . . 10 |- ( x e. { B } -> x = B )
3	1, 2	syl6 33	. . . . . . . . 9 |- ( A C_ { B } -> ( x e. A -> x = B ) )
4		eleq1 2240	. . . . . . . . 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 1819	. . . . . 6 |- ( A C_ { B } -> ( E. x x e. A -> B e. A ) )
8		snssi 3738	. . . . . 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 3172	. . . 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 3211	. 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 1353   E.wex 1492   e. wcel 2148   C_ wss 3131   {csn 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-sn 3600

This theorem is referenced by:  eqsnm  3757  ss1o0el1  4199  exmidn0m  4203  exmidsssn  4204  exmidomni  7142  exmidunben  12429  exmidsbthrlem  14809  sbthomlem  14812