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Mirrors > Home > MPE Home > Th. List > ordtr1 | Structured version Visualization version 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 5654 | . 2 ⊢ (Ord 𝐶 → Tr 𝐶) | |
2 | trel 4687 | . 2 ⊢ (Tr 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Tr wtr 4680 Ord word 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-uni 4373 df-tr 4681 df-ord 5643 |
This theorem is referenced by: ontr1 5688 dfsmo2 7331 smores2 7338 smoel 7344 smogt 7351 ordiso2 8303 r1ordg 8524 r1pwss 8530 r1val1 8532 rankr1ai 8544 rankval3b 8572 rankonidlem 8574 onssr1 8577 cofsmo 8974 fpwwe2lem9 9339 bnj1098 30108 bnj594 30236 |
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