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| Mirrors > Home > MPE Home > Th. List > ontri1 | Structured version Visualization version GIF version | ||
| Description: A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.) |
| Ref | Expression |
|---|---|
| ontri1 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eloni 6394 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 2 | eloni 6394 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
| 3 | ordtri1 6417 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | |
| 4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 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: oneqmini 6436 onmindif 6476 onint 7810 onnmin 7818 onmindif2 7827 dfom2 7889 ondif2 8540 oaword 8587 oawordeulem 8592 oaf1o 8601 odi 8617 omeulem1 8620 oeeulem 8639 oeeui 8640 nnmword 8671 cofonr 8712 naddel1 8725 naddss1 8727 domtriord 9163 sdomel 9164 onsdominel 9166 ordunifi 9326 cantnfp1lem3 9720 oemapvali 9724 cantnflem1b 9726 cantnflem1 9729 cnfcom3lem 9743 rankr1clem 9860 rankelb 9864 rankval3b 9866 rankr1a 9876 unbndrank 9882 rankxplim3 9921 cardne 10005 carden2b 10007 cardsdomel 10014 carddom2 10017 harcard 10018 domtri2 10029 infxpenlem 10053 alephord 10115 alephord3 10118 alephle 10128 dfac12k 10188 cflim2 10303 cofsmo 10309 cfsmolem 10310 isf32lem5 10397 pwcfsdom 10623 pwfseqlem3 10700 inar1 10815 om2uzlt2i 13992 sltval2 27701 sltres 27707 nosepssdm 27731 nolt02olem 27739 nolt02o 27740 nogt01o 27741 noetasuplem4 27781 noetainflem4 27785 nocvxminlem 27822 madebdaylemlrcut 27937 om2noseqlt2 28306 nummin 35105 onsuct0 36442 onint1 36450 onmaxnelsup 43235 onsupnmax 43240 onsupuni 43241 oninfint 43248 onsupmaxb 43251 onsupeqnmax 43259 oe0suclim 43290 cantnfresb 43337 cantnf2 43338 tfsconcatfv 43354 tfsnfin 43365 oadif1lem 43392 oadif1 43393 naddwordnexlem4 43414 ontric3g 43535 infordmin 43545 minregex 43547 alephiso3 43572 |
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