Theorem ordin 4432

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 4425	. . 3 |- ( Ord A -> Tr A )
2		ordtr 4425	. . 3 |- ( Ord B -> Tr B )
3		trin 4152	. . 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 3394	. . 3 |- ( A i^i B ) C_ B
6		trssord 4427	. . 3 |- ( ( Tr ( A i^i B ) /\ ( A i^i B ) C_ B /\ Ord B ) -> Ord ( A i^i B ) )
7	5, 6	mp3an2 1338	. 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 3165   C_ wss 3166   Tr wtr 4142   Ord word 4409

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413

This theorem is referenced by:  onin  4433  smores  6378  smores2  6380