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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 6194 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ord0eln0 6238 | . 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 207 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 Ord word 6183 Oncon0 6184 |
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 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: ondif1 8115 oe0lem 8127 oevn0 8129 oa00 8174 omord 8183 om00 8190 om00el 8191 omeulem1 8197 omeulem2 8198 oewordri 8207 oeordsuc 8209 oelim2 8210 oeoa 8212 oeoe 8214 oeeui 8217 omabs 8263 omxpenlem 8606 cantnff 9125 cantnfp1 9132 cantnflem1d 9139 cantnflem1 9140 cantnflem3 9142 cantnflem4 9143 cantnf 9144 cnfcomlem 9150 cnfcom3 9155 r1tskina 10192 onsucconni 33682 onint1 33694 frlmpwfi 39576 |
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