Theorem trssord 2216

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 2188	. . . . 5 |- (A (_ B -> (E We B -> E We A))
2	1	imp 277	. . . 4 |- ((A (_ B /\ E We B) -> E We A)
3		ordwe 2212	. . . 4 |- (Ord B -> E We B)
4	2, 3	sylan2 346	. . 3 |- ((A (_ B /\ Ord B) -> E We A)
5	4	anim2i 270	. 2 |- ((Tr A /\ (A (_ B /\ Ord B)) -> (Tr A /\ E We A))
6		3anass 585	. 2 |- ((Tr A /\ A (_ B /\ Ord B) <-> (Tr A /\ (A (_ B /\ Ord B)))
7		df-ord 2202	. 2 |- (Ord A <-> (Tr A /\ E We A))
8	5, 6, 7	3imtr4 192	1 |- ((Tr A /\ A (_ B /\ Ord B) -> Ord A)

Syntax hints:   -> wi 2   /\ wa 196   /\ w3a 581   (_ wss 1487  Tr wtr 2041  Ecep 2056   We wwe 2062  Ord word 2198

This theorem is referenced by:  ordin 2228  ssorduni 2249  suceloni 2314  ordom 2382  tfrlem8 2956  ondomon 3662

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202

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