Theorem snsssn 3792

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 3172	. . 3 |- ( { A } C_ { B } <-> A. x ( x e. { A } -> x e. { B } ) )
2		velsn 3640	. . . . 5 |- ( x e. { A } <-> x = A )
3		velsn 3640	. . . . 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 1484	. . 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 3045	. . 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 1362   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   {csn 3623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990  df-in 3163  df-ss 3170  df-sn 3629

This theorem is referenced by: (None)