Theorem onin 2229

Description: The intersection of two ordinal numbers is an ordinal number.

Assertion

Ref	Expression
onin	⊢ ((A ∈ On ∧ B ∈ On) → (A ∩ B) ∈ On)

Proof of Theorem onin

Step	Hyp	Ref	Expression
1		ordin 2228	. . 3 ⊢ ((Ord A ∧ Ord B) → Ord (A ∩ B))
2		eloni 2209	. . 3 ⊢ (A ∈ On → Ord A)
3		eloni 2209	. . 3 ⊢ (B ∈ On → Ord B)
4	1, 2, 3	syl2an 349	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → Ord (A ∩ B))
5		pm3.26 256	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → A ∈ On)
6		inex1g 1699	. . 3 ⊢ (A ∈ On → (A ∩ B) ∈ V)
7		elong 2207	. . 3 ⊢ ((A ∩ B) ∈ V → ((A ∩ B) ∈ On ↔ Ord (A ∩ B)))
8	5, 6, 7	3syl 21	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → ((A ∩ B) ∈ On ↔ Ord (A ∩ B)))
9	4, 8	mpbird 171	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∩ B) ∈ On)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486  Ord word 2198  Oncon0 2199

This theorem is referenced by:  tfrlem5 2953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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