Theorem ordin 4482

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 4475	. . 3 |- ( Ord A -> Tr A )
2		ordtr 4475	. . 3 |- ( Ord B -> Tr B )
3		trin 4197	. . 3 |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )
4	1, 2, 3	syl2an 289	. 2 |- ( ( Ord A /\ Ord B ) -> Tr ( A i^i B ) )
5		inss2 3428	. . 3 |- ( A i^i B ) C_ B
6		trssord 4477	. . 3 |- ( ( Tr ( A i^i B ) /\ ( A i^i B ) C_ B /\ Ord B ) -> Ord ( A i^i B ) )
7	5, 6	mp3an2 1361	. 2 |- ( ( Tr ( A i^i B ) /\ Ord B ) -> Ord ( A i^i B ) )
8	4, 7	sylancom 420	1 |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   i^i cin 3199   C_ wss 3200   Tr wtr 4187   Ord word 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463

This theorem is referenced by:  onin  4483  smores  6457  smores2  6459