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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 4411 | . 2 ⊢ Ord 𝐴 |
3 | ordtr 4363 | . 2 ⊢ (Ord 𝐴 → Tr 𝐴) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ Tr 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 Tr wtr 4087 Ord word 4347 Oncon0 4348 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 |
This theorem is referenced by: onunisuci 4417 exmidonfinlem 7170 bj-el2oss1o 13809 nnsf 14038 |
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