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Theorem onin 4089
 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 4078 . . 3 (A On → Ord A)
2 eloni 4078 . . 3 (B On → Ord B)
3 ordin 4088 . . 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 3884 . . 3 (A On → (AB) V)
7 elong 4076 . . 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 1390  Vcvv 2551   ∩ cin 2910  Ord word 4065  Oncon0 4066 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071 This theorem is referenced by:  tfrlem5  5871
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