MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trin2 Unicode version

Theorem trin2 5395
Description: The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
trin2

Proof of Theorem trin2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cotr 5384 . . . 4
2 cotr 5384 . . . . . 6
3 brin 4501 . . . . . . . . . . . . 13
4 brin 4501 . . . . . . . . . . . . 13
5 simpr 461 . . . . . . . . . . . . . . . 16
6 simpl 457 . . . . . . . . . . . . . . . 16
75, 6anim12d 563 . . . . . . . . . . . . . . 15
87com12 31 . . . . . . . . . . . . . 14
98an4s 826 . . . . . . . . . . . . 13
103, 4, 9syl2anb 479 . . . . . . . . . . . 12
1110com12 31 . . . . . . . . . . 11
12 brin 4501 . . . . . . . . . . 11
1311, 12syl6ibr 227 . . . . . . . . . 10
1413alanimi 1637 . . . . . . . . 9
1514alanimi 1637 . . . . . . . 8
1615alanimi 1637 . . . . . . 7
1716ex 434 . . . . . 6
182, 17sylbi 195 . . . . 5
1918com12 31 . . . 4
201, 19sylbi 195 . . 3
2120imp 429 . 2
22 cotr 5384 . 2
2321, 22sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  i^icin 3474  C_wss 3475   class class class wbr 4452  o.ccom 5008
This theorem is referenced by:  trinxp  5397  trficl  37779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-co 5013
  Copyright terms: Public domain W3C validator