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Theorem cotr 5939
 Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 5938. (Contributed by NM, 27-Dec-1996.)
Assertion
Ref Expression
cotr ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable group:   𝑥,𝑦,𝑧,𝑅

Proof of Theorem cotr
StepHypRef Expression
1 cotrg 5938 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   ⊆ wss 3881   class class class wbr 5030   ∘ ccom 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-co 5528 This theorem is referenced by:  xpidtr  5949  trin2  5950  dfer2  8273  trclfvcotr  14360  pslem  17808  letsr  17829  dirtr  17838  filnetlem3  33838  dftrrels3  35969  dftrrel3  35971  dfeqvrels3  35981  cotrintab  40309  iunrelexpuztr  40415
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