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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 4284 | . 2 ⊢ 𝒫 {∅} = {∅, {∅}} | |
2 | p0ex 4779 | . . 3 ⊢ {∅} ∈ V | |
3 | 2 | pwex 4774 | . 2 ⊢ 𝒫 {∅} ∈ V |
4 | 1, 3 | eqeltrri 2685 | 1 ⊢ {∅, {∅}} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-pr 4128 |
This theorem is referenced by: ord3ex 4782 zfpair 4831 |
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