Theorem sssnm 3741

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 3141	. . . . . . . . . 10 |- ( A C_ { B } -> ( x e. A -> x e. { B } ) )
2		elsni 3601	. . . . . . . . . 10 |- ( x e. { B } -> x = B )
3	1, 2	syl6 33	. . . . . . . . 9 |- ( A C_ { B } -> ( x e. A -> x = B ) )
4		eleq1 2233	. . . . . . . . 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 1812	. . . . . 6 |- ( A C_ { B } -> ( E. x x e. A -> B e. A ) )
8		snssi 3724	. . . . . 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 3162	. . . 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 3201	. 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 1348   E.wex 1485   e. wcel 2141   C_ wss 3121   {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589

This theorem is referenced by:  eqsnm  3742  ss1o0el1  4183  exmidn0m  4187  exmidsssn  4188  exmidomni  7118  exmidunben  12381  exmidsbthrlem  14054  sbthomlem  14057