Theorem pwpw0ss 3928

Description: Compute the power set of the power set of the empty set. (See pw0 3860 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	⊢ {∅, {∅}} ⊆ 𝒫 {∅}

Proof of Theorem pwpw0ss

Step	Hyp	Ref	Expression
1		pwsnss 3927	1 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}

Syntax hints:   ⊆ wss 3220  ∅c0 3520  𝒫 cpw 3688  {csn 3708  {cpr 3709

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715

This theorem is referenced by:  pp0ex  4324  exmidpw  7209  exmidpweq  7210  pw1dom2  7580  pw1ne1  7582