Theorem pwpw0ss 3831

Description: Compute the power set of the power set of the empty set. (See pw0 3766 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 3830	1 |- { (/) , { (/) } } C_ ~P { (/) }

Syntax hints:   C_ wss 3154   (/)c0 3447   ~Pcpw 3602   {csn 3619   {cpr 3620

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626

This theorem is referenced by:  pp0ex  4219  exmidpw  6966  exmidpweq  6967  pw1dom2  7289  pw1ne1  7291