Theorem pwpwpw0 4863

Description: Compute the power set of the power set of the power set of the empty set. (See also pw0 4773 and pwpw0 4774.) (Contributed by NM, 2-May-2009.)

Assertion

Ref	Expression
pwpwpw0	⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})

Proof of Theorem pwpwpw0

Step	Hyp	Ref	Expression
1		pwpr 4861	1 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})

Syntax hints:   = wceq 1563   ∪ cun 3905  ∅c0 4288  𝒫 cpw 4558  {csn 4585  {cpr 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-pw 4560  df-sn 4586  df-pr 4588

This theorem is referenced by: (None)