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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 4265 | . 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 1465 Ord word 4254 Oncon0 4255 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 |
This theorem is referenced by: elon2 4268 onelon 4276 onin 4278 onelss 4279 ontr1 4281 onordi 4318 onss 4379 suceloni 4387 sucelon 4389 onsucmin 4393 onsucelsucr 4394 onintonm 4403 ordsucunielexmid 4416 onsucuni2 4449 nnord 4495 tfrlem1 6173 tfrlemisucaccv 6190 tfrlemibfn 6193 tfrlemiubacc 6195 tfrexlem 6199 tfr1onlemsucfn 6205 tfr1onlemsucaccv 6206 tfr1onlembfn 6209 tfr1onlemubacc 6211 tfrcllemsucfn 6218 tfrcllemsucaccv 6219 tfrcllembfn 6222 tfrcllemubacc 6224 sucinc2 6310 phplem4on 6729 ordiso 6889 |
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