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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 4350 | . 2 ⊢ (Ord 𝐶 → Tr 𝐶) | |
2 | trel 4081 | . 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 2135 Tr wtr 4074 Ord word 4334 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 df-uni 3784 df-tr 4075 df-iord 4338 |
This theorem is referenced by: ontr1 4361 ordwe 4547 dfsmo2 6246 smores2 6253 smoel 6259 tfr1onlemsucaccv 6300 tfr1onlembxssdm 6302 tfr1onlembfn 6303 tfr1onlemaccex 6307 tfr1onlemres 6308 tfrcllemsucaccv 6313 tfrcllembxssdm 6315 tfrcllembfn 6316 tfrcllemaccex 6320 tfrcllemres 6321 tfrcl 6323 ordiso2 6991 |
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