Theorem ordin 6393

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 6376	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		ordtr 6376	. . 3 ⊢ (Ord 𝐵 → Tr 𝐵)
3		trin 5223	. . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 608	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵))
5		inss2 4134	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
6		trssord 6379	. . 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   ⊆ wss 3895   ∩ cin 4089  Tr wtr 5211  Ord word 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-uni 4867  df-tr 5212  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365

This theorem is used by:  onin  6394  ordtri3or  6395  ordelinel  6466  smores  8364  smores2  8366  ordtypelem5  9520  ordtypelem7  9522