|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 9155 | . . 3 ⊢ ¬ 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴 | |
| 2 | canth2g 9172 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) | |
| 3 | pwexg 5377 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
| 4 | canth2g 9172 | . . . . . 6 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) | |
| 5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) | 
| 6 | sdomtr 9156 | . . . . 5 ⊢ ((𝐴 ≺ 𝒫 𝐴 ∧ 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) → 𝐴 ≺ 𝒫 𝒫 𝐴) | |
| 7 | 2, 5, 6 | syl2anc 584 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝒫 𝐴) | 
| 8 | breq1 5145 | . . . 4 ⊢ (𝒫 𝒫 𝐴 = 𝐴 → (𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴 ↔ 𝐴 ≺ 𝒫 𝒫 𝐴)) | |
| 9 | 7, 8 | syl5ibrcom 247 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝒫 𝐴 = 𝐴 → 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)) | 
| 10 | 1, 9 | mtoi 199 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝒫 𝐴 = 𝐴) | 
| 11 | 10 | neqned 2946 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ≠ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 𝒫 cpw 4599 class class class wbr 5142 ≺ csdm 8985 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |