Theorem snsspw 3794

Description: The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
snsspw	|- { A } C_ ~P A

Proof of Theorem snsspw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqimss 3237	. . 3 |- ( x = A -> x C_ A )
2		velsn 3639	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3607	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2307	. . 3 |- ( x e. ~P A <-> x C_ A )
5	1, 2, 4	3imtr4i 201	. 2 |- ( x e. { A } -> x e. ~P A )
6	5	ssriv 3187	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1364   e. wcel 2167   C_ wss 3157   ~Pcpw 3605   {csn 3622

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-in 3163  df-ss 3170  df-pw 3607  df-sn 3628

This theorem is referenced by:  snexg  4217