Theorem pwsnss 3924

Description: The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)

Assertion

Ref	Expression
pwsnss	|- { (/) , { A } } C_ ~P { A }

Proof of Theorem pwsnss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sssnr 3873	. . 3 |- ( ( x = (/) \/ x = { A } ) -> x C_ { A } )
2	1	ss2abi 3320	. 2 |- { x | ( x = (/) \/ x = { A } ) } C_ { x | x C_ { A } }
3		dfpr2 3724	. 2 |- { (/) , { A } } = { x | ( x = (/) \/ x = { A } ) }
4		df-pw 3687	. 2 |- ~P { A } = { x | x C_ { A } }
5	2, 3, 4	3sstr4i 3289	1 |- { (/) , { A } } C_ ~P { A }

Syntax hints:   \/ wo 720   = wceq 1402   {cab 2224   C_ wss 3220   (/)c0 3520   ~Pcpw 3685   {csn 3705   {cpr 3706

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712

This theorem is referenced by:  pwpw0ss  3925