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Mirrors > Home > ILE Home > Th. List > onordi | GIF version |
Description: An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
onordi | ⊢ Ord 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . 2 ⊢ 𝐴 ∈ On | |
2 | eloni 4393 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ Ord 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 Ord word 4380 Oncon0 4381 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-in 3150 df-ss 3157 df-uni 3825 df-tr 4117 df-iord 4384 df-on 4386 |
This theorem is referenced by: ontrci 4445 onsucssi 4523 onsucsssucexmid 4544 onirri 4560 |
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