Theorem ordin 6386

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 6369	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		ordtr 6369	. . 3 ⊢ (Ord 𝐵 → Tr 𝐵)
3		trin 5224	. . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 608	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵))
5		inss2 4183	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
6		trssord 6372	. . 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 3898   ⊆ wss 3899  Tr wtr 5212  Ord word 6354

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 2147  ax-9 2155  ax-ext 2733

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-in 3906  df-ss 3916  df-uni 4868  df-tr 5213  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6358

This theorem is used by:  onin  6387  ordtri3or  6388  ordelinel  6459  smores  8344  smores2  8346  ordtypelem5  9500  ordtypelem7  9502