Theorem snsssn 3654

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 3052	. . 3 |- ( { A } C_ { B } <-> A. x ( x e. { A } -> x e. { B } ) )
2		velsn 3510	. . . . 5 |- ( x e. { A } <-> x = A )
3		velsn 3510	. . . . 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 1429	. . 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 2932	. . 3 |- ( A e. _V -> ( A. x ( x = A -> x = B ) -> A = B ) )
9	7, 8	ax-mp 7	. 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 1312   = wceq 1314   e. wcel 1463   _Vcvv 2657   C_ wss 3037   {csn 3493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879  df-in 3043  df-ss 3050  df-sn 3499

This theorem is referenced by: (None)