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| Mirrors > Home > MPE Home > Th. List > cotr | Structured version Visualization version GIF version | ||
| Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 6127. (Contributed by NM, 27-Dec-1996.) |
| Ref | Expression |
|---|---|
| cotr | ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cotrg 6127 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ⊆ wss 3951 class class class wbr 5143 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-co 5694 |
| This theorem is referenced by: xpidtr 6142 trin2 6143 dfer2 8746 trclfvcotr 15048 pslem 18617 letsr 18638 dirtr 18647 filnetlem3 36381 dftrrels3 38577 dftrrel3 38579 dfeqvrels3 38590 cotrintab 43627 iunrelexpuztr 43732 |
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