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Mirrors > Home > ILE Home > Th. List > ontrci | GIF version |
Description: An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
ontrci | ⊢ Tr 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . . 3 ⊢ 𝐴 ∈ On | |
2 | 1 | onordi 4348 | . 2 ⊢ Ord 𝐴 |
3 | ordtr 4300 | . 2 ⊢ (Ord 𝐴 → Tr 𝐴) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ Tr 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Tr wtr 4026 Ord word 4284 Oncon0 4285 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-in 3077 df-ss 3084 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 |
This theorem is referenced by: onunisuci 4354 exmidonfinlem 7049 bj-el2oss1o 12991 nnsf 13209 |
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