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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 4827. (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 5241 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴) | |
2 | eqimss 4020 | . . 3 ⊢ (𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴) | |
3 | 2 | necon3bi 3039 | . 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 2105 ≠ wne 3013 ⊆ wss 3933 𝒫 cpw 4535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-pw 4537 |
This theorem is referenced by: pnfnemnf 10684 |
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