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Mirrors > Home > MPE Home > Th. List > 2pwne | Structured version Visualization version GIF version |
Description: No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.) |
Ref | Expression |
---|---|
2pwne | ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomirr 8653 | . . 3 ⊢ ¬ 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴 | |
2 | canth2g 8670 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) | |
3 | pwexg 5278 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
4 | canth2g 8670 | . . . . . 6 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) |
6 | sdomtr 8654 | . . . . 5 ⊢ ((𝐴 ≺ 𝒫 𝐴 ∧ 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) → 𝐴 ≺ 𝒫 𝒫 𝐴) | |
7 | 2, 5, 6 | syl2anc 586 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝒫 𝐴) |
8 | breq1 5068 | . . . 4 ⊢ (𝒫 𝒫 𝐴 = 𝐴 → (𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴 ↔ 𝐴 ≺ 𝒫 𝒫 𝐴)) | |
9 | 7, 8 | syl5ibrcom 249 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝒫 𝐴 = 𝐴 → 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)) |
10 | 1, 9 | mtoi 201 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝒫 𝐴 = 𝐴) |
11 | 10 | neqned 3023 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 𝒫 cpw 4538 class class class wbr 5065 ≺ csdm 8507 |
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 ax-pr 5329 ax-un 7460 |
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-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 |
This theorem is referenced by: (None) |
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