Theorem snsspw 3870

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 3294	. . 3 |- ( x = A -> x C_ A )
2		velsn 3708	. . 3 |- ( x e. { A } <-> x = A )
3		df-pw 3673	. . . 4 |- ~P A = { x | x C_ A }
4	3	abeq2i 2345	. . 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 3244	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1398   e. wcel 2205   C_ wss 3213   ~Pcpw 3671   {csn 3691

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697

This theorem is referenced by:  snexg  4299