Theorem ordin 3689

Description: The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37.

Assertion

Ref	Expression
ordin	|- ((Ord A /\ Ord B) -> Ord (A i^i B))

Proof of Theorem ordin

Step	Hyp	Ref	Expression
1		trin 3422	. . 3 |- ((Tr A /\ Tr B) -> Tr (A i^i B))
2		ordtr 3672	. . 3 |- (Ord A -> Tr A)
3		ordtr 3672	. . 3 |- (Ord B -> Tr B)
4	1, 2, 3	syl2an 503	. 2 |- ((Ord A /\ Ord B) -> Tr (A i^i B))
5		inss2 2813	. . 3 |- (A i^i B) C_ B
6		trssord 3675	. . 3 |- ((Tr (A i^i B) /\ (A i^i B) C_ B /\ Ord B) -> Ord (A i^i B))
7	5, 6	mp3an2 1179	. 2 |- ((Tr (A i^i B) /\ Ord B) -> Ord (A i^i B))
8	4, 7	sylancom 531	1 |- ((Ord A /\ Ord B) -> Ord (A i^i B))

Syntax hints:   -> wi 3   /\ wa 240   i^i cin 2592   C_ wss 2593  Tr wtr 3411  Ord word 3656

This theorem is referenced by:  onin 3690  ordtri3or 3691  ordtri3orOLD 3692  smores 16446

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-uni 3178  df-tr 3412  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660

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