ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  trin2 Structured version   GIF version

Theorem trin2 4659
Description: The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
trin2 (((𝑅𝑅) ⊆ 𝑅 (𝑆𝑆) ⊆ 𝑆) → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ (𝑅𝑆))

Proof of Theorem trin2
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cotr 4649 . . . 4 ((𝑅𝑅) ⊆ 𝑅xyz((x𝑅y y𝑅z) → x𝑅z))
2 cotr 4649 . . . . . 6 ((𝑆𝑆) ⊆ 𝑆xyz((x𝑆y y𝑆z) → x𝑆z))
3 brin 3802 . . . . . . . . . . . . 13 (x(𝑅𝑆)y ↔ (x𝑅y x𝑆y))
4 brin 3802 . . . . . . . . . . . . 13 (y(𝑅𝑆)z ↔ (y𝑅z y𝑆z))
5 simpr 103 . . . . . . . . . . . . . . . 16 ((((x𝑆y y𝑆z) → x𝑆z) ((x𝑅y y𝑅z) → x𝑅z)) → ((x𝑅y y𝑅z) → x𝑅z))
6 simpl 102 . . . . . . . . . . . . . . . 16 ((((x𝑆y y𝑆z) → x𝑆z) ((x𝑅y y𝑅z) → x𝑅z)) → ((x𝑆y y𝑆z) → x𝑆z))
75, 6anim12d 318 . . . . . . . . . . . . . . 15 ((((x𝑆y y𝑆z) → x𝑆z) ((x𝑅y y𝑅z) → x𝑅z)) → (((x𝑅y y𝑅z) (x𝑆y y𝑆z)) → (x𝑅z x𝑆z)))
87com12 27 . . . . . . . . . . . . . 14 (((x𝑅y y𝑅z) (x𝑆y y𝑆z)) → ((((x𝑆y y𝑆z) → x𝑆z) ((x𝑅y y𝑅z) → x𝑅z)) → (x𝑅z x𝑆z)))
98an4s 522 . . . . . . . . . . . . 13 (((x𝑅y x𝑆y) (y𝑅z y𝑆z)) → ((((x𝑆y y𝑆z) → x𝑆z) ((x𝑅y y𝑅z) → x𝑅z)) → (x𝑅z x𝑆z)))
103, 4, 9syl2anb 275 . . . . . . . . . . . 12 ((x(𝑅𝑆)y y(𝑅𝑆)z) → ((((x𝑆y y𝑆z) → x𝑆z) ((x𝑅y y𝑅z) → x𝑅z)) → (x𝑅z x𝑆z)))
1110com12 27 . . . . . . . . . . 11 ((((x𝑆y y𝑆z) → x𝑆z) ((x𝑅y y𝑅z) → x𝑅z)) → ((x(𝑅𝑆)y y(𝑅𝑆)z) → (x𝑅z x𝑆z)))
12 brin 3802 . . . . . . . . . . 11 (x(𝑅𝑆)z ↔ (x𝑅z x𝑆z))
1311, 12syl6ibr 151 . . . . . . . . . 10 ((((x𝑆y y𝑆z) → x𝑆z) ((x𝑅y y𝑅z) → x𝑅z)) → ((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z))
1413alanimi 1345 . . . . . . . . 9 ((z((x𝑆y y𝑆z) → x𝑆z) z((x𝑅y y𝑅z) → x𝑅z)) → z((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z))
1514alanimi 1345 . . . . . . . 8 ((yz((x𝑆y y𝑆z) → x𝑆z) yz((x𝑅y y𝑅z) → x𝑅z)) → yz((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z))
1615alanimi 1345 . . . . . . 7 ((xyz((x𝑆y y𝑆z) → x𝑆z) xyz((x𝑅y y𝑅z) → x𝑅z)) → xyz((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z))
1716ex 108 . . . . . 6 (xyz((x𝑆y y𝑆z) → x𝑆z) → (xyz((x𝑅y y𝑅z) → x𝑅z) → xyz((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z)))
182, 17sylbi 114 . . . . 5 ((𝑆𝑆) ⊆ 𝑆 → (xyz((x𝑅y y𝑅z) → x𝑅z) → xyz((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z)))
1918com12 27 . . . 4 (xyz((x𝑅y y𝑅z) → x𝑅z) → ((𝑆𝑆) ⊆ 𝑆xyz((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z)))
201, 19sylbi 114 . . 3 ((𝑅𝑅) ⊆ 𝑅 → ((𝑆𝑆) ⊆ 𝑆xyz((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z)))
2120imp 115 . 2 (((𝑅𝑅) ⊆ 𝑅 (𝑆𝑆) ⊆ 𝑆) → xyz((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z))
22 cotr 4649 . 2 (((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ (𝑅𝑆) ↔ xyz((x(𝑅𝑆)y y(𝑅𝑆)z) → x(𝑅𝑆)z))
2321, 22sylibr 137 1 (((𝑅𝑅) ⊆ 𝑅 (𝑆𝑆) ⊆ 𝑆) → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ (𝑅𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240  cin 2910  wss 2911   class class class wbr 3755  ccom 4292
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:  trinxp  4661
  Copyright terms: Public domain W3C validator