Theorem pwpwpw0 4570

Description: Compute the power set of the power set of the power set of the empty set. (See also pw0 4478 and pwpw0 4479.) (Contributed by NM, 2-May-2009.)

Assertion

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

Proof of Theorem pwpwpw0

Step	Hyp	Ref	Expression
1		pwpr 4568	1 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})

Syntax hints:   = wceq 1631   ∪ cun 3721  ∅c0 4063  𝒫 cpw 4297  {csn 4316  {cpr 4318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319

This theorem is referenced by: (None)