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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 4295 | . 2 ⊢ (Ord 𝐶 → Tr 𝐶) | |
2 | trel 4028 | . 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 103 ∈ wcel 1480 Tr wtr 4021 Ord word 4279 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-uni 3732 df-tr 4022 df-iord 4283 |
This theorem is referenced by: ontr1 4306 ordwe 4485 dfsmo2 6177 smores2 6184 smoel 6190 tfr1onlemsucaccv 6231 tfr1onlembxssdm 6233 tfr1onlembfn 6234 tfr1onlemaccex 6238 tfr1onlemres 6239 tfrcllemsucaccv 6244 tfrcllembxssdm 6246 tfrcllembfn 6247 tfrcllemaccex 6251 tfrcllemres 6252 tfrcl 6254 ordiso2 6913 |
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