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Theorem onin 4072
Description: The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
onin ((A On B On) → (AB) On)

Proof of Theorem onin
StepHypRef Expression
1 eloni 4061 . . 3 (A On → Ord A)
2 eloni 4061 . . 3 (B On → Ord B)
3 ordin 4071 . . 3 ((Ord A Ord B) → Ord (AB))
41, 2, 3syl2an 273 . 2 ((A On B On) → Ord (AB))
5 simpl 102 . . 3 ((A On B On) → A On)
6 inex1g 3867 . . 3 (A On → (AB) V)
7 elong 4059 . . 3 ((AB) V → ((AB) On ↔ Ord (AB)))
85, 6, 73syl 17 . 2 ((A On B On) → ((AB) On ↔ Ord (AB)))
94, 8mpbird 156 1 ((A On B On) → (AB) On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1374  Vcvv 2535  cin 2893  Ord word 4048  Oncon0 4049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054
This theorem is referenced by:  tfrlem5  5852
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