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Theorem onin 6415
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 6394 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6394 . . 3 (𝐵 ∈ On → Ord 𝐵)
3 ordin 6414 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
41, 2, 3syl2an 596 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴𝐵))
5 simpl 482 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6 inex1g 5319 . . 3 (𝐴 ∈ On → (𝐴𝐵) ∈ V)
7 elong 6392 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
85, 6, 73syl 18 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
94, 8mpbird 257 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  Vcvv 3480  cin 3950  Ord word 6383  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-uni 4908  df-tr 5260  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  tfrlem5  8420  noreson  27705  ontopbas  36429
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