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Theorem onin 6399
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 6378 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6378 . . 3 (𝐵 ∈ On → Ord 𝐵)
3 ordin 6398 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
41, 2, 3syl2an 594 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴𝐵))
5 simpl 481 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6 inex1g 5316 . . 3 (𝐴 ∈ On → (𝐴𝐵) ∈ V)
7 elong 6376 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
85, 6, 73syl 18 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
94, 8mpbird 256 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2099  Vcvv 3462  cin 3945  Ord word 6367  Oncon0 6368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rab 3420  df-v 3464  df-in 3953  df-ss 3963  df-uni 4906  df-tr 5263  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-ord 6371  df-on 6372
This theorem is referenced by:  tfrlem5  8402  noreson  27687  ontopbas  36153
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