Theorem snsssn 3763

Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Hypothesis

Ref	Expression
sneqr.1	|- A e. _V

Assertion

Ref	Expression
snsssn	|- ( { A } C_ { B } -> A = B )

Proof of Theorem snsssn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3146	. . 3 |- ( { A } C_ { B } <-> A. x ( x e. { A } -> x e. { B } ) )
2		velsn 3611	. . . . 5 |- ( x e. { A } <-> x = A )
3		velsn 3611	. . . . 5 |- ( x e. { B } <-> x = B )
4	2, 3	imbi12i 239	. . . 4 |- ( ( x e. { A } -> x e. { B } ) <-> ( x = A -> x = B ) )
5	4	albii 1470	. . 3 |- ( A. x ( x e. { A } -> x e. { B } ) <-> A. x ( x = A -> x = B ) )
6	1, 5	bitri 184	. 2 |- ( { A } C_ { B } <-> A. x ( x = A -> x = B ) )
7		sneqr.1	. . 3 |- A e. _V
8		sbceqal 3020	. . 3 |- ( A e. _V -> ( A. x ( x = A -> x = B ) -> A = B ) )
9	7, 8	ax-mp 5	. 2 |- ( A. x ( x = A -> x = B ) -> A = B )
10	6, 9	sylbi 121	1 |- ( { A } C_ { B } -> A = B )

Syntax hints:   -> wi 4   A.wal 1351   = wceq 1353   e. wcel 2148   _Vcvv 2739   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-sbc 2965  df-in 3137  df-ss 3144  df-sn 3600

This theorem is referenced by: (None)