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Mirrors > Home > MPE Home > Th. List > pwne0 | Structured version Visualization version GIF version |
Description: A power class is never empty. (Contributed by NM, 3-Sep-2018.) |
Ref | Expression |
---|---|
pwne0 | ⊢ 𝒫 𝐴 ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 4760 | . 2 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | 1 | ne0ii 3882 | 1 ⊢ 𝒫 𝐴 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2780 ∅c0 3874 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-nul 4717 |
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-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 |
This theorem is referenced by: undefne0 7292 |
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