Theorem cotr 6111

Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 6108. (Contributed by NM, 27-Dec-1996.)

Assertion

Ref	Expression
cotr	⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Distinct variable group:   𝑥,𝑦,𝑧,𝑅

Proof of Theorem cotr

Step	Hyp	Ref	Expression
1		cotrg 6108	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1532   ⊆ wss 3945   class class class wbr 5143   ∘ ccom 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-xp 5679  df-rel 5680  df-co 5682

This theorem is referenced by:  xpidtr  6123  trin2  6124  dfer2  8720  trclfvcotr  14983  pslem  18558  letsr  18579  dirtr  18588  filnetlem3  35859  dftrrels3  38043  dftrrel3  38045  dfeqvrels3  38056  cotrintab  43035  iunrelexpuztr  43140