Theorem pwsnss 3799

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 3749	. . 3 |- ( ( x = (/) \/ x = { A } ) -> x C_ { A } )
2	1	ss2abi 3225	. 2 |- { x | ( x = (/) \/ x = { A } ) } C_ { x | x C_ { A } }
3		dfpr2 3608	. 2 |- { (/) , { A } } = { x | ( x = (/) \/ x = { A } ) }
4		df-pw 3574	. 2 |- ~P { A } = { x | x C_ { A } }
5	2, 3, 4	3sstr4i 3194	1 |- { (/) , { A } } C_ ~P { A }

Syntax hints:   \/ wo 708   = wceq 1353   {cab 2161   C_ wss 3127   (/)c0 3420   ~Pcpw 3572   {csn 3589   {cpr 3590

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596

This theorem is referenced by:  pwpw0ss  3800