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Theorem onin 4914
Description: The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
onin

Proof of Theorem onin
StepHypRef Expression
1 eloni 4893 . . 3
2 eloni 4893 . . 3
3 ordin 4913 . . 3
41, 2, 3syl2an 477 . 2
5 simpl 457 . . 3
6 inex1g 4595 . . 3
7 elong 4891 . . 3
85, 6, 73syl 20 . 2
94, 8mpbird 232 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818   cvv 3109  i^icin 3474  Ordword 4882   con0 4883
This theorem is referenced by:  tfrlem5  7068  noreson  29420  ontopbas  29893
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  ax-sep 4573
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  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-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-uni 4250  df-tr 4546  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887
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