Theorem snsspw 3744

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 3196	. . 3 |- ( x = A -> x C_ A )
2		velsn 3593	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3561	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2277	. . 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 3146	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1343   e. wcel 2136   C_ wss 3116   ~Pcpw 3559   {csn 3576

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582

This theorem is referenced by:  snexg  4163