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| Mirrors > Home > MPE Home > Th. List > pp0ex | Structured version Visualization version GIF version | ||
| Description: The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.) |
| Ref | Expression |
|---|---|
| pp0ex | ⊢ {∅, {∅}} ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pwpw0 4813 | . 2 ⊢ 𝒫 {∅} = {∅, {∅}} | |
| 2 | p0ex 5384 | . . 3 ⊢ {∅} ∈ V | |
| 3 | 2 | pwex 5380 | . 2 ⊢ 𝒫 {∅} ∈ V |
| 4 | 1, 3 | eqeltrri 2838 | 1 ⊢ {∅, {∅}} ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: ord3ex 5387 zfpair 5421 |
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