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Mirrors > Home > ILE Home > Th. List > ordtr1 | GIF version |
Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) |
Ref | Expression |
---|---|
ordtr1 | ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 4133 | . 2 ⊢ (Ord 𝐶 → Tr 𝐶) | |
2 | trel 3882 | . 2 ⊢ (Tr 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 Tr wtr 3875 Ord word 4117 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-in 2979 df-ss 2986 df-uni 3602 df-tr 3876 df-iord 4121 |
This theorem is referenced by: ontr1 4144 ordwe 4318 dfsmo2 5925 smores2 5932 smoel 5938 ordiso2 6446 |
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