Theorem trssord 2960

Description: A transitive subclass of an ordinal class is ordinal.

Assertion

Ref	Expression
trssord	|- ((Tr A /\ A (_ B /\ Ord B) -> Ord A)

Proof of Theorem trssord

Step	Hyp	Ref	Expression
1		wess 2931	. . . . 5 |- (A (_ B -> (E We B -> E We A))
2	1	imp 350	. . . 4 |- ((A (_ B /\ E We B) -> E We A)
3		ordwe 2956	. . . 4 |- (Ord B -> E We B)
4	2, 3	sylan2 451	. . 3 |- ((A (_ B /\ Ord B) -> E We A)
5	4	anim2i 335	. 2 |- ((Tr A /\ (A (_ B /\ Ord B)) -> (Tr A /\ E We A))
6		3anass 778	. 2 |- ((Tr A /\ A (_ B /\ Ord B) <-> (Tr A /\ (A (_ B /\ Ord B)))
7		df-ord 2946	. 2 |- (Ord A <-> (Tr A /\ E We A))
8	5, 6, 7	3imtr4 219	1 |- ((Tr A /\ A (_ B /\ Ord B) -> Ord A)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   (_ wss 2043  Tr wtr 2675  Ecep 2825   We wwe 2911  Ord word 2942

This theorem is referenced by:  ordin 2972  ssorduni 2988  suceloni 3057  ordom 3136  ondomon 4836

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-br 2615  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946

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