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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 8448 | . . 3 ⊢ ¬ 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴 | |
2 | canth2g 8465 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) | |
3 | pwexg 5128 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
4 | canth2g 8465 | . . . . . 6 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) |
6 | sdomtr 8449 | . . . . 5 ⊢ ((𝐴 ≺ 𝒫 𝐴 ∧ 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) → 𝐴 ≺ 𝒫 𝒫 𝐴) | |
7 | 2, 5, 6 | syl2anc 576 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝒫 𝐴) |
8 | breq1 4928 | . . . 4 ⊢ (𝒫 𝒫 𝐴 = 𝐴 → (𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴 ↔ 𝐴 ≺ 𝒫 𝒫 𝐴)) | |
9 | 7, 8 | syl5ibrcom 239 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝒫 𝐴 = 𝐴 → 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)) |
10 | 1, 9 | mtoi 191 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝒫 𝐴 = 𝐴) |
11 | 10 | neqned 2967 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 Vcvv 3408 𝒫 cpw 4416 class class class wbr 4925 ≺ csdm 8303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 |
This theorem is referenced by: (None) |
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