ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cotr GIF version

Theorem cotr 4878
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) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable group:   𝑥,𝑦,𝑧,𝑅

Proof of Theorem cotr
StepHypRef Expression
1 df-co 4508 . . . 4 (𝑅𝑅) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧)}
21relopabi 4625 . . 3 Rel (𝑅𝑅)
3 ssrel 4587 . . 3 (Rel (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
42, 3ax-mp 7 . 2 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
5 vex 2660 . . . . . . . 8 𝑥 ∈ V
6 vex 2660 . . . . . . . 8 𝑧 ∈ V
75, 6opelco 4671 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧))
8 df-br 3896 . . . . . . . 8 (𝑥𝑅𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝑅)
98bicomi 131 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝑅𝑥𝑅𝑧)
107, 9imbi12i 238 . . . . . 6 ((⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11 19.23v 1837 . . . . . 6 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1210, 11bitr4i 186 . . . . 5 ((⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1312albii 1429 . . . 4 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
14 alcom 1437 . . . 4 (∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1513, 14bitri 183 . . 3 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1615albii 1429 . 2 (∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
174, 16bitri 183 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312  wex 1451  wcel 1463  wss 3037  cop 3496   class class class wbr 3895  ccom 4503  Rel wrel 4504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-co 4508
This theorem is referenced by:  xpidtr  4887  trin2  4888  dfer2  6384
  Copyright terms: Public domain W3C validator