Theorem snsssn 3741

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 3131	. . 3 |- ( { A } C_ { B } <-> A. x ( x e. { A } -> x e. { B } ) )
2		velsn 3593	. . . . 5 |- ( x e. { A } <-> x = A )
3		velsn 3593	. . . . 5 |- ( x e. { B } <-> x = B )
4	2, 3	imbi12i 238	. . . 4 |- ( ( x e. { A } -> x e. { B } ) <-> ( x = A -> x = B ) )
5	4	albii 1458	. . 3 |- ( A. x ( x e. { A } -> x e. { B } ) <-> A. x ( x = A -> x = B ) )
6	1, 5	bitri 183	. 2 |- ( { A } C_ { B } <-> A. x ( x = A -> x = B ) )
7		sneqr.1	. . 3 |- A e. _V
8		sbceqal 3006	. . 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 120	1 |- ( { A } C_ { B } -> A = B )

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   {csn 3576

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-in 3122  df-ss 3129  df-sn 3582

This theorem is referenced by: (None)