Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4388 | . 2 ⊢ Ord 𝐴 |
3 | ordtr 4340 | . 2 ⊢ (Ord 𝐴 → Tr 𝐴) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ Tr 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 Tr wtr 4064 Ord word 4324 Oncon0 4325 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-in 3108 df-ss 3115 df-uni 3775 df-tr 4065 df-iord 4328 df-on 4330 |
This theorem is referenced by: onunisuci 4394 exmidonfinlem 7130 bj-el2oss1o 13419 nnsf 13648 |
Copyright terms: Public domain | W3C validator |