|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > pwpwpw0 | Structured version Visualization version GIF version | ||
| Description: Compute the power set of the power set of the power set of the empty set. (See also pw0 4811 and pwpw0 4812.) (Contributed by NM, 2-May-2009.) | 
| Ref | Expression | 
|---|---|
| pwpwpw0 | ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pwpr 4900 | 1 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∪ cun 3948 ∅c0 4332 𝒫 cpw 4599 {csn 4625 {cpr 4627 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-pw 4601 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |