Theorem pwsnss 3833

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 3783	. . 3 |- ( ( x = (/) \/ x = { A } ) -> x C_ { A } )
2	1	ss2abi 3255	. 2 |- { x | ( x = (/) \/ x = { A } ) } C_ { x | x C_ { A } }
3		dfpr2 3641	. 2 |- { (/) , { A } } = { x | ( x = (/) \/ x = { A } ) }
4		df-pw 3607	. 2 |- ~P { A } = { x | x C_ { A } }
5	2, 3, 4	3sstr4i 3224	1 |- { (/) , { A } } C_ ~P { A }

Syntax hints:   \/ wo 709   = wceq 1364   {cab 2182   C_ wss 3157   (/)c0 3450   ~Pcpw 3605   {csn 3622   {cpr 3623

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629

This theorem is referenced by:  pwpw0ss  3834