Theorem snsspw 4375

Description: The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-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 3657	. . 3 |- ( x = A -> x C_ A )
2		velsn 4193	. . 3 |- ( x e. { A } <-> x = A )
3		selpw 4165	. . 3 |- ( x e. ~P A <-> x C_ A )
4	1, 2, 3	3imtr4i 281	. 2 |- ( x e. { A } -> x e. ~P A )
5	4	ssriv 3607	1 |- { A } C_ ~P A

Syntax hints:   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-sn 4178

This theorem is referenced by:  snexALT  4852  snwf  8672  tsksn  9582