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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 4356 | . 2 ⊢ Ord 𝐴 |
3 | ordtr 4308 | . 2 ⊢ (Ord 𝐴 → Tr 𝐴) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ Tr 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 Tr wtr 4034 Ord word 4292 Oncon0 4293 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-in 3082 df-ss 3089 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 |
This theorem is referenced by: onunisuci 4362 exmidonfinlem 7066 bj-el2oss1o 13152 nnsf 13374 |
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