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Theorem onin 4268
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 4257 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 eloni 4257 . . 3 (𝐵 ∈ On → Ord 𝐵)
3 ordin 4267 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
41, 2, 3syl2an 285 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴𝐵))
5 simpl 108 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6 inex1g 4024 . . 3 (𝐴 ∈ On → (𝐴𝐵) ∈ V)
7 elong 4255 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
85, 6, 73syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) ∈ On ↔ Ord (𝐴𝐵)))
94, 8mpbird 166 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1463  Vcvv 2657  cin 3036  Ord word 4244  Oncon0 4245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-in 3043  df-ss 3050  df-uni 3703  df-tr 3987  df-iord 4248  df-on 4250
This theorem is referenced by:  tfrlem5  6165
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