Theorem snsspw 3776

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 3221	. . 3 |- ( x = A -> x C_ A )
2		velsn 3621	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3589	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2298	. . 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 3171	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1363   e. wcel 2158   C_ wss 3141   ~Pcpw 3587   {csn 3604

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610

This theorem is referenced by:  snexg  4196