Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  onin GIF version

Theorem onin 4169
 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 4158 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 eloni 4158 . . 3 (𝐵 ∈ On → Ord 𝐵)
3 ordin 4168 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
41, 2, 3syl2an 283 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴𝐵))
5 simpl 107 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6 inex1g 3934 . . 3 (𝐴 ∈ On → (𝐴𝐵) ∈ V)
7 elong 4156 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
85, 6, 73syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
94, 8mpbird 165 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∈ wcel 1434  Vcvv 2610   ∩ cin 2981  Ord word 4145  Oncon0 4146 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-in 2988  df-ss 2995  df-uni 3622  df-tr 3896  df-iord 4149  df-on 4151 This theorem is referenced by:  tfrlem5  5983
 Copyright terms: Public domain W3C validator