Theorem pwsnss 3790

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 3740	. . 3 |- ( ( x = (/) \/ x = { A } ) -> x C_ { A } )
2	1	ss2abi 3219	. 2 |- { x | ( x = (/) \/ x = { A } ) } C_ { x | x C_ { A } }
3		dfpr2 3602	. 2 |- { (/) , { A } } = { x | ( x = (/) \/ x = { A } ) }
4		df-pw 3568	. 2 |- ~P { A } = { x | x C_ { A } }
5	2, 3, 4	3sstr4i 3188	1 |- { (/) , { A } } C_ ~P { A }

Syntax hints:   \/ wo 703   = wceq 1348   {cab 2156   C_ wss 3121   (/)c0 3414   ~Pcpw 3566   {csn 3583   {cpr 3584

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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590

This theorem is referenced by:  pwpw0ss  3791