Theorem ordin 2972

Description: The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37.

Assertion

Ref	Expression
ordin	|- ((Ord A /\ Ord B) -> Ord (A i^i B))

Proof of Theorem ordin

Step	Hyp	Ref	Expression
1		trin 2685	. . 3 |- ((Tr A /\ Tr B) -> Tr (A i^i B))
2		ordtr 2957	. . 3 |- (Ord A -> Tr A)
3		ordtr 2957	. . 3 |- (Ord B -> Tr B)
4	1, 2, 3	syl2an 454	. 2 |- ((Ord A /\ Ord B) -> Tr (A i^i B))
5		inss2 2227	. . 3 |- (A i^i B) (_ B
6		trssord 2960	. . 3 |- ((Tr (A i^i B) /\ (A i^i B) (_ B /\ Ord B) -> Ord (A i^i B))
7	5, 6	mp3an2 902	. 2 |- ((Tr (A i^i B) /\ Ord B) -> Ord (A i^i B))
8	4, 7	sylancom 475	1 |- ((Ord A /\ Ord B) -> Ord (A i^i B))

Syntax hints:   -> wi 3   /\ wa 223   i^i cin 2042   (_ wss 2043  Tr wtr 2675  Ord word 2942

This theorem is referenced by:  onin 2973  ordtri3or 2974

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-tr 2676  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946

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