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Theorem onin 4064
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 4053 . . 3 (A On → Ord A)
2 eloni 4053 . . 3 (B On → Ord B)
3 ordin 4063 . . 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 3859 . . 3 (A On → (AB) V)
7 elong 4051 . . 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 1369  Vcvv 2529  cin 2887  Ord word 4040  Oncon0 4041
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 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998  ax-sep 3841
This theorem depends on definitions:  df-bi 110  df-3an 871  df-tru 1229  df-nf 1326  df-sb 1622  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-ral 2283  df-rex 2284  df-v 2531  df-in 2895  df-ss 2902  df-uni 3547  df-tr 3821  df-iord 4044  df-on 4046
This theorem is referenced by:  tfrlem5  5843
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