Theorem ordin 6382

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 6366	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		ordtr 6366	. . 3 ⊢ (Ord 𝐵 → Tr 𝐵)
3		trin 5241	. . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 596	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵))
5		inss2 4213	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
6		trssord 6369	. . 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 588	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∩ cin 3925   ⊆ wss 3926  Tr wtr 5229  Ord word 6351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rab 3416  df-v 3461  df-in 3933  df-ss 3943  df-uni 4884  df-tr 5230  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-ord 6355

This theorem is referenced by:  onin  6383  ordtri3or  6384  ordelinel  6455  smores  8366  smores2  8368  ordtypelem5  9536  ordtypelem7  9538