Theorem pwpw0ss 3893

Description: Compute the power set of the power set of the empty set. (See pw0 3825 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 3892	1 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}

Syntax hints:   ⊆ wss 3201  ∅c0 3496  𝒫 cpw 3656  {csn 3673  {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680

This theorem is referenced by:  pp0ex  4285  exmidpw  7143  exmidpweq  7144  pw1dom2  7488  pw1ne1  7490