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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 4385 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ On ↔ Ord 𝐴)) | |
2 | 1 | ibi 176 | 1 ⊢ (𝐴 ∈ On → Ord 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 Ord word 4374 Oncon0 4375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-in 3147 df-ss 3154 df-uni 3822 df-tr 4114 df-iord 4378 df-on 4380 |
This theorem is referenced by: elon2 4388 onelon 4396 onin 4398 onelss 4399 ontr1 4401 onordi 4438 onss 4504 onsuc 4512 onsucb 4514 onsucmin 4518 onsucelsucr 4519 onintonm 4528 ordsucunielexmid 4542 onsucuni2 4575 nnord 4623 tfrlem1 6323 tfrlemisucaccv 6340 tfrlemibfn 6343 tfrlemiubacc 6345 tfrexlem 6349 tfr1onlemsucfn 6355 tfr1onlemsucaccv 6356 tfr1onlembfn 6359 tfr1onlemubacc 6361 tfrcllemsucfn 6368 tfrcllemsucaccv 6369 tfrcllembfn 6372 tfrcllemubacc 6374 sucinc2 6461 phplem4on 6881 ordiso 7049 |
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