Theorem snsspw 3847

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 3281	. . 3 |- ( x = A -> x C_ A )
2		velsn 3686	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3654	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2342	. . 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 3231	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1397   e. wcel 2202   C_ wss 3200   ~Pcpw 3652   {csn 3669

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675

This theorem is referenced by:  snexg  4274