Theorem pwpwpw0 4859

Description: Compute the power set of the power set of the power set of the empty set. (See also pw0 4768 and pwpw0 4769.) (Contributed by NM, 2-May-2009.)

Assertion

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

Proof of Theorem pwpwpw0

Step	Hyp	Ref	Expression
1		pwpr 4857	1 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})

Syntax hints:   = wceq 1541   ∪ cun 3899  ∅c0 4285  𝒫 cpw 4554  {csn 4580  {cpr 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-pw 4556  df-sn 4581  df-pr 4583

This theorem is referenced by: (None)