Theorem snsspw 3751

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 3201	. . 3 |- ( x = A -> x C_ A )
2		velsn 3600	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3568	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2281	. . 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 3151	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1348   e. wcel 2141   C_ wss 3121   ~Pcpw 3566   {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by:  snexg  4170