Theorem pwpw0ss 3731

Description: Compute the power set of the power set of the empty set. (See pw0 3667 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 3730	1 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}

Syntax hints:   ⊆ wss 3071  ∅c0 3363  𝒫 cpw 3510  {csn 3527  {cpr 3528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534

This theorem is referenced by:  pp0ex  4113  exmidpw  6802  pw1dom2  13190