Theorem trssord 2971

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 2942	. . . . 5 |- (A (_ B -> (E We B -> E We A))
2	1	imp 350	. . . 4 |- ((A (_ B /\ E We B) -> E We A)
3		ordwe 2967	. . . 4 |- (Ord B -> E We B)
4	2, 3	sylan2 453	. . 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 781	. 2 |- ((Tr A /\ A (_ B /\ Ord B) <-> (Tr A /\ (A (_ B /\ Ord B)))
7		df-ord 2957	. 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 777   (_ wss 2050  Tr wtr 2685  Ecep 2836   We wwe 2922  Ord word 2953

This theorem is referenced by:  ordin 2983  ssorduni 2999  suceloni 3068  ordom 3147  ondomon 4867

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957

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