Theorem snsssn 3802

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		ssalel 3181	. . 3 |- ( { A } C_ { B } <-> A. x ( x e. { A } -> x e. { B } ) )
2		velsn 3650	. . . . 5 |- ( x e. { A } <-> x = A )
3		velsn 3650	. . . . 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 1493	. . 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 3054	. . 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 1371   = wceq 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999  df-in 3172  df-ss 3179  df-sn 3639

This theorem is referenced by: (None)