Theorem ordin 4384

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 4377	. . 3 |- ( Ord A -> Tr A )
2		ordtr 4377	. . 3 |- ( Ord B -> Tr B )
3		trin 4110	. . 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 3356	. . 3 |- ( A i^i B ) C_ B
6		trssord 4379	. . 3 |- ( ( Tr ( A i^i B ) /\ ( A i^i B ) C_ B /\ Ord B ) -> Ord ( A i^i B ) )
7	5, 6	mp3an2 1325	. 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 3128   C_ wss 3129   Tr wtr 4100   Ord word 4361

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-uni 3810  df-tr 4101  df-iord 4365

This theorem is referenced by:  onin  4385  smores  6289  smores2  6291