Theorem trssord 2965

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 2936	. . . . 5 |- (A (_ B -> (E We B -> E We A))
2	1	imp 350	. . . 4 |- ((A (_ B /\ E We B) -> E We A)
3		ordwe 2961	. . . 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 779	. 2 |- ((Tr A /\ A (_ B /\ Ord B) <-> (Tr A /\ (A (_ B /\ Ord B)))
7		df-ord 2951	. 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 775   (_ wss 2047  Tr wtr 2680  Ecep 2830   We wwe 2916  Ord word 2947

This theorem is referenced by:  ordin 2977  ssorduni 2993  suceloni 3062  ordom 3141  ondomon 4856

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951

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