Theorem ordin 6221

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 6205	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		ordtr 6205	. . 3 ⊢ (Ord 𝐵 → Tr 𝐵)
3		trin 5156	. . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 599	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵))
5		inss2 4130	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
6		trssord 6208	. . 3 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
7	5, 6	mp3an2 1451	. 2 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
8	4, 7	sylancom 591	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∩ cin 3852   ⊆ wss 3853  Tr wtr 5146  Ord word 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-11 2160  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rab 3060  df-v 3400  df-in 3860  df-ss 3870  df-uni 4806  df-tr 5147  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-ord 6194

This theorem is referenced by:  onin  6222  ordtri3or  6223  ordelinel  6289  smores  8067  smores2  8069  ordtypelem5  9116  ordtypelem7  9118