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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 4211 | . 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 1439 Ord word 4200 Oncon0 4201 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2624 df-in 3008 df-ss 3015 df-uni 3662 df-tr 3945 df-iord 4204 df-on 4206 |
This theorem is referenced by: elon2 4214 onelon 4222 onin 4224 onelss 4225 ontr1 4227 onordi 4264 onss 4325 suceloni 4333 sucelon 4335 onsucmin 4339 onsucelsucr 4340 onintonm 4349 ordsucunielexmid 4362 onsucuni2 4395 nnord 4441 tfrlem1 6089 tfrlemisucaccv 6106 tfrlemibfn 6109 tfrlemiubacc 6111 tfrexlem 6115 tfr1onlemsucfn 6121 tfr1onlemsucaccv 6122 tfr1onlembfn 6125 tfr1onlemubacc 6127 tfrcllemsucfn 6134 tfrcllemsucaccv 6135 tfrcllembfn 6138 tfrcllemubacc 6140 sucinc2 6223 phplem4on 6639 ordiso 6785 |
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