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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 4371. (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 4756 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴) | |
2 | eqimss 3620 | . . 3 ⊢ (𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴) | |
3 | 2 | necon3bi 2808 | . 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 1977 ≠ wne 2780 ⊆ wss 3540 𝒫 cpw 4108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-rab 2905 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 |
This theorem is referenced by: pnfnemnf 9973 |
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