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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 4745 | . 2 ⊢ 𝒫 {∅} = {∅, {∅}} | |
2 | p0ex 5284 | . . 3 ⊢ {∅} ∈ V | |
3 | 2 | pwex 5280 | . 2 ⊢ 𝒫 {∅} ∈ V |
4 | 1, 3 | eqeltrri 2910 | 1 ⊢ {∅, {∅}} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3494 ∅c0 4290 𝒫 cpw 4538 {csn 4566 {cpr 4568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-pw 4540 df-sn 4567 df-pr 4569 |
This theorem is referenced by: ord3ex 5287 zfpair 5321 |
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