Theorem pwpw0ss 3835

Description: Compute the power set of the power set of the empty set. (See pw0 3770 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with subset in place of equality). (Contributed by Jim Kingdon, 12-Aug-2018.)

Assertion

Ref	Expression
pwpw0ss	|- { (/) , { (/) } } C_ ~P { (/) }

Proof of Theorem pwpw0ss

Step	Hyp	Ref	Expression
1		pwsnss 3834	1 |- { (/) , { (/) } } C_ ~P { (/) }

Syntax hints:   C_ wss 3157   (/)c0 3451   ~Pcpw 3606   {csn 3623   {cpr 3624

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 3452  df-pw 3608  df-sn 3629  df-pr 3630

This theorem is referenced by:  pp0ex  4223  exmidpw  6978  exmidpweq  6979  pw1dom2  7310  pw1ne1  7312