Theorem pwpw0ss 3882

Description: Compute the power set of the power set of the empty set. (See pw0 3814 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 3881	1 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}

Syntax hints:   ⊆ wss 3197  ∅c0 3491  𝒫 cpw 3649  {csn 3666  {cpr 3667

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673

This theorem is referenced by:  pp0ex  4272  exmidpw  7058  exmidpweq  7059  pw1dom2  7400  pw1ne1  7402