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Mirrors > Home > MPE Home > Th. List > on0eln0 | Structured version Visualization version GIF version |
Description: An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.) |
Ref | Expression |
---|---|
on0eln0 | ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 5650 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ord0eln0 5696 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 Ord word 5639 Oncon0 5640 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: ondif1 7468 oe0lem 7480 oevn0 7482 oa00 7526 omord 7535 om00 7542 om00el 7543 omeulem1 7549 omeulem2 7550 oewordri 7559 oeordsuc 7561 oelim2 7562 oeoa 7564 oeoe 7566 oeeui 7569 omabs 7614 omxpenlem 7946 cantnff 8454 cantnfp1lem2 8459 cantnfp1lem3 8460 cantnfp1 8461 cantnflem1d 8468 cantnflem1 8469 cantnflem3 8471 cantnflem4 8472 cantnf 8473 cnfcomlem 8479 cnfcom3 8484 r1tskina 9483 onsucconi 31606 onint1 31618 frlmpwfi 36686 |
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