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Theorem trin 4555
 Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
Assertion
Ref Expression
trin

Proof of Theorem trin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3686 . . . . 5
2 trss 4554 . . . . . 6
3 trss 4554 . . . . . 6
42, 3im2anan9 835 . . . . 5
51, 4syl5bi 217 . . . 4
6 ssin 3719 . . . 4
75, 6syl6ib 226 . . 3
87ralrimiv 2869 . 2
9 dftr3 4549 . 2
108, 9sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  A.wral 2807  i^icin 3474  C_wss 3475  Trwtr 4545 This theorem is referenced by:  ordin  4913  tcmin  8193  ingru  9214  gruina  9217  dfon2lem4  29218 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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3482  df-ss 3489  df-uni 4250  df-tr 4546
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