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Theorem trin2 5249
 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 5238 . . . 4
2 cotr 5238 . . . . . 6
3 brin 4251 . . . . . . . . . . . . 13
4 brin 4251 . . . . . . . . . . . . 13
5 simpr 448 . . . . . . . . . . . . . . . 16
6 simpl 444 . . . . . . . . . . . . . . . 16
75, 6anim12d 547 . . . . . . . . . . . . . . 15
87com12 29 . . . . . . . . . . . . . 14
98an4s 800 . . . . . . . . . . . . 13
103, 4, 9syl2anb 466 . . . . . . . . . . . 12
1110com12 29 . . . . . . . . . . 11
12 brin 4251 . . . . . . . . . . 11
1311, 12syl6ibr 219 . . . . . . . . . 10
1413alanimi 1571 . . . . . . . . 9
1514alanimi 1571 . . . . . . . 8
1615alanimi 1571 . . . . . . 7
1716ex 424 . . . . . 6
182, 17sylbi 188 . . . . 5
1918com12 29 . . . 4
201, 19sylbi 188 . . 3
2120imp 419 . 2
22 cotr 5238 . 2
2321, 22sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549   cin 3311   wss 3312   class class class wbr 4204   ccom 4874 This theorem is referenced by:  trinxp  5251 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-co 4879
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