Theorem snsspw 3699

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 3156	. . 3 |- ( x = A -> x C_ A )
2		velsn 3549	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3517	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2251	. . 3 |- ( x e. ~P A <-> x C_ A )
5	1, 2, 4	3imtr4i 200	. 2 |- ( x e. { A } -> x e. ~P A )
6	5	ssriv 3106	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1332   e. wcel 1481   C_ wss 3076   ~Pcpw 3515   {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538

This theorem is referenced by:  snexg  4116