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Theorem cotr 4649
Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cotr ((𝑅𝑅) ⊆ 𝑅xyz((x𝑅y y𝑅z) → x𝑅z))
Distinct variable group:   x,y,z,𝑅

Proof of Theorem cotr
StepHypRef Expression
1 df-co 4297 . . . 4 (𝑅𝑅) = {⟨x, z⟩ ∣ y(x𝑅y y𝑅z)}
21relopabi 4406 . . 3 Rel (𝑅𝑅)
3 ssrel 4371 . . 3 (Rel (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅xz(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅)))
42, 3ax-mp 7 . 2 ((𝑅𝑅) ⊆ 𝑅xz(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅))
5 vex 2554 . . . . . . . 8 x V
6 vex 2554 . . . . . . . 8 z V
75, 6opelco 4450 . . . . . . 7 (⟨x, z (𝑅𝑅) ↔ y(x𝑅y y𝑅z))
8 df-br 3756 . . . . . . . 8 (x𝑅z ↔ ⟨x, z 𝑅)
98bicomi 123 . . . . . . 7 (⟨x, z 𝑅x𝑅z)
107, 9imbi12i 228 . . . . . 6 ((⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ (y(x𝑅y y𝑅z) → x𝑅z))
11 19.23v 1760 . . . . . 6 (y((x𝑅y y𝑅z) → x𝑅z) ↔ (y(x𝑅y y𝑅z) → x𝑅z))
1210, 11bitr4i 176 . . . . 5 ((⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ y((x𝑅y y𝑅z) → x𝑅z))
1312albii 1356 . . . 4 (z(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ zy((x𝑅y y𝑅z) → x𝑅z))
14 alcom 1364 . . . 4 (zy((x𝑅y y𝑅z) → x𝑅z) ↔ yz((x𝑅y y𝑅z) → x𝑅z))
1513, 14bitri 173 . . 3 (z(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ yz((x𝑅y y𝑅z) → x𝑅z))
1615albii 1356 . 2 (xz(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ xyz((x𝑅y y𝑅z) → x𝑅z))
174, 16bitri 173 1 ((𝑅𝑅) ⊆ 𝑅xyz((x𝑅y y𝑅z) → x𝑅z))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378   wcel 1390  wss 2911  cop 3370   class class class wbr 3755  ccom 4292  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-co 4297
This theorem is referenced by:  xpidtr  4658  trin2  4659  dfer2  6043
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