Theorem cotr 6006

Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 6005. (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 6005	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   ⊆ wss 3883   class class class wbr 5070   ∘ ccom 5584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-co 5589

This theorem is referenced by:  xpidtr  6016  trin2  6017  dfer2  8457  trclfvcotr  14648  pslem  18205  letsr  18226  dirtr  18235  filnetlem3  34496  dftrrels3  36617  dftrrel3  36619  dfeqvrels3  36629  cotrintab  41111  iunrelexpuztr  41216