Theorem pwsnss 3805

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 3755	. . 3 |- ( ( x = (/) \/ x = { A } ) -> x C_ { A } )
2	1	ss2abi 3229	. 2 |- { x | ( x = (/) \/ x = { A } ) } C_ { x | x C_ { A } }
3		dfpr2 3613	. 2 |- { (/) , { A } } = { x | ( x = (/) \/ x = { A } ) }
4		df-pw 3579	. 2 |- ~P { A } = { x | x C_ { A } }
5	2, 3, 4	3sstr4i 3198	1 |- { (/) , { A } } C_ ~P { A }

Syntax hints:   \/ wo 708   = wceq 1353   {cab 2163   C_ wss 3131   (/)c0 3424   ~Pcpw 3577   {csn 3594   {cpr 3595

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601

This theorem is referenced by:  pwpw0ss  3806