Theorem ordin 6398

Description: The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)

Assertion

Ref	Expression
ordin	⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))

Proof of Theorem ordin

Step	Hyp	Ref	Expression
1		ordtr 6381	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		ordtr 6381	. . 3 ⊢ (Ord 𝐵 → Tr 𝐵)
3		trin 5235	. . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 608	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵))
5		inss2 4193	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
6		trssord 6384	. . 3 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
7	5, 6	mp3an2 1478	. 2 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
8	4, 7	sylancom 600	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∩ cin 3907   ⊆ wss 3908  Tr wtr 5223  Ord word 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rab 3420  df-v 3460  df-in 3915  df-ss 3925  df-uni 4878  df-tr 5224  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6370

This theorem is used by:  onin  6399  ordtri3or  6400  ordelinel  6471  smores  8348  smores2  8350  ordtypelem5  9494  ordtypelem7  9496