Theorem ordin 4420

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 4413	. . 3 |- ( Ord A -> Tr A )
2		ordtr 4413	. . 3 |- ( Ord B -> Tr B )
3		trin 4141	. . 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 3384	. . 3 |- ( A i^i B ) C_ B
6		trssord 4415	. . 3 |- ( ( Tr ( A i^i B ) /\ ( A i^i B ) C_ B /\ Ord B ) -> Ord ( A i^i B ) )
7	5, 6	mp3an2 1336	. 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 3156   C_ wss 3157   Tr wtr 4131   Ord word 4397

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401

This theorem is referenced by:  onin  4421  smores  6350  smores2  6352