Theorem cotr 5647

Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 5646. (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 5646	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1629   ⊆ wss 3723   class class class wbr 4786   ∘ ccom 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-co 5258

This theorem is referenced by:  xpidtr  5657  trin2  5658  dfer2  7897  trclfvcotr  13954  pslem  17410  letsr  17431  dirtr  17440  filnetlem3  32708  cotrintab  38444  iunrelexpuztr  38534