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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 4270 | . 2 ⊢ (Ord 𝐶 → Tr 𝐶) | |
2 | trel 4003 | . 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 1465 Tr wtr 3996 Ord word 4254 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 |
This theorem is referenced by: ontr1 4281 ordwe 4460 dfsmo2 6152 smores2 6159 smoel 6165 tfr1onlemsucaccv 6206 tfr1onlembxssdm 6208 tfr1onlembfn 6209 tfr1onlemaccex 6213 tfr1onlemres 6214 tfrcllemsucaccv 6219 tfrcllembxssdm 6221 tfrcllembfn 6222 tfrcllemaccex 6226 tfrcllemres 6227 tfrcl 6229 ordiso2 6888 |
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