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Theorem trin 4500
 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 3608 . . . . 5
2 trss 4499 . . . . . 6
3 trss 4499 . . . . . 6
42, 3im2anan9 853 . . . . 5
51, 4syl5bi 225 . . . 4
6 ssin 3645 . . . 4
75, 6syl6ib 234 . . 3
87ralrimiv 2808 . 2
9 dftr3 4494 . 2
108, 9sylibr 217 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wcel 1904  wral 2756   cin 3389   wss 3390   wtr 4490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-in 3397  df-ss 3404  df-uni 4191  df-tr 4491 This theorem is referenced by:  ordin  5460  tcmin  8243  ingru  9258  gruina  9261  dfon2lem4  30503
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