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Mirrors > Home > MPE Home > Th. List > pwne | Structured version Visualization version GIF version |
Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4909. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
pwne | ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnss 5358 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴) | |
2 | eqimss 4054 | . . 3 ⊢ (𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴) | |
3 | 2 | necon3bi 2965 | . 2 ⊢ (¬ 𝒫 𝐴 ⊆ 𝐴 → 𝒫 𝐴 ≠ 𝐴) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 𝒫 cpw 4605 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-pw 4607 |
This theorem is referenced by: pnfnemnf 11314 |
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