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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 6394 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 2 | ord0eln0 6439 | . 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 206 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: ondif1 8539 oe0lem 8551 oevn0 8553 oa00 8597 omord 8606 om00 8613 om00el 8614 omeulem1 8620 omeulem2 8621 oewordri 8630 oeordsuc 8632 oelim2 8633 oeoa 8635 oeoe 8637 oeeui 8640 omabs 8689 omxpenlem 9113 cantnff 9714 cantnfp1 9721 cantnflem1d 9728 cantnflem1 9729 cantnflem3 9731 cantnflem4 9732 cantnf 9733 cnfcomlem 9739 cnfcom3 9744 r1tskina 10822 onsucconni 36438 onint1 36450 frlmpwfi 43110 omge1 43310 omge2 43311 omlim2 43312 omord2lim 43313 omord2i 43314 dflim5 43342 tfsconcatb0 43357 tfsconcat0b 43359 oaun3lem1 43387 naddwordnexlem4 43414 omltoe 43420 |
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