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Theorem cotr 5414
Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 5413. (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 5413 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wss 3539   class class class wbr 4577  ccom 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-rel 5035  df-co 5037
This theorem is referenced by:  xpidtr  5424  trin2  5425  dfer2  7607  trclfvcotr  13544  pslem  16975  letsr  16996  dirtr  17005  filnetlem3  31351  cotrintab  36736  iunrelexpuztr  36826
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