Theorem snsspw 3790

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 3233	. . 3 |- ( x = A -> x C_ A )
2		velsn 3635	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3603	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2304	. . 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 3183	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1364   e. wcel 2164   C_ wss 3153   ~Pcpw 3601   {csn 3618

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624

This theorem is referenced by:  snexg  4213