Theorem pwpw0ss 3888

Description: Compute the power set of the power set of the empty set. (See pw0 3820 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 3887	1 |- { (/) , { (/) } } C_ ~P { (/) }

Syntax hints:   C_ wss 3200   (/)c0 3494   ~Pcpw 3652   {csn 3669   {cpr 3670

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676

This theorem is referenced by:  pp0ex  4279  exmidpw  7099  exmidpweq  7100  pw1dom2  7444  pw1ne1  7446