Theorem snsspw 3727

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 3182	. . 3 |- ( x = A -> x C_ A )
2		velsn 3577	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3545	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2268	. . 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 3132	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1335   e. wcel 2128   C_ wss 3102   ~Pcpw 3543   {csn 3560

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566

This theorem is referenced by:  snexg  4145