Theorem ordin 4140

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 i^i B ) )

Proof of Theorem ordin

Step	Hyp	Ref	Expression
1		ordtr 4133	. . 3 |- ( Ord A -> Tr A )
2		ordtr 4133	. . 3 |- ( Ord B -> Tr B )
3		trin 3885	. . 3 |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )
4	1, 2, 3	syl2an 283	. 2 |- ( ( Ord A /\ Ord B ) -> Tr ( A i^i B ) )
5		inss2 3187	. . 3 |- ( A i^i B ) C_ B
6		trssord 4135	. . 3 |- ( ( Tr ( A i^i B ) /\ ( A i^i B ) C_ B /\ Ord B ) -> Ord ( A i^i B ) )
7	5, 6	mp3an2 1256	. 2 |- ( ( Tr ( A i^i B ) /\ Ord B ) -> Ord ( A i^i B ) )
8	4, 7	sylancom 411	1 |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 102   i^i cin 2972   C_ wss 2973   Tr wtr 3875   Ord word 4117

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121

This theorem is referenced by:  onin  4141  smores  5930  smores2  5932