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Theorem onin 6190
 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 6169 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6169 . . 3 (𝐵 ∈ On → Ord 𝐵)
3 ordin 6189 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
41, 2, 3syl2an 598 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴𝐵))
5 simpl 486 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6 inex1g 5187 . . 3 (𝐴 ∈ On → (𝐴𝐵) ∈ V)
7 elong 6167 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
85, 6, 73syl 18 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
94, 8mpbird 260 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880  Ord word 6158  Oncon0 6159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2770  ax-sep 5167 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-tr 5137  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163 This theorem is referenced by:  tfrlem5  7999  noreson  33277  ontopbas  33886
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