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Mirrors > Home > ILE Home > Th. List > eloni | GIF version |
Description: An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.) |
Ref | Expression |
---|---|
eloni | ⊢ (𝐴 ∈ On → Ord 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elong 4346 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴)) | |
2 | 1 | ibi 175 | 1 ⊢ (𝐴 ∈ On → Ord 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 Ord word 4335 Oncon0 4336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-in 3118 df-ss 3125 df-uni 3785 df-tr 4076 df-iord 4339 df-on 4341 |
This theorem is referenced by: elon2 4349 onelon 4357 onin 4359 onelss 4360 ontr1 4362 onordi 4399 onss 4465 suceloni 4473 sucelon 4475 onsucmin 4479 onsucelsucr 4480 onintonm 4489 ordsucunielexmid 4503 onsucuni2 4536 nnord 4584 tfrlem1 6268 tfrlemisucaccv 6285 tfrlemibfn 6288 tfrlemiubacc 6290 tfrexlem 6294 tfr1onlemsucfn 6300 tfr1onlemsucaccv 6301 tfr1onlembfn 6304 tfr1onlemubacc 6306 tfrcllemsucfn 6313 tfrcllemsucaccv 6314 tfrcllembfn 6317 tfrcllemubacc 6319 sucinc2 6406 phplem4on 6825 ordiso 6993 |
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