Theorem trssord 5740

Description: A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)

Assertion

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

Proof of Theorem trssord

Step	Hyp	Ref	Expression
1		wess 5101	. . . . 5 |- ( A C_ B -> ( _E We B -> _E We A ) )
2		ordwe 5736	. . . . 5 |- ( Ord B -> _E We B )
3	1, 2	impel 485	. . . 4 |- ( ( A C_ B /\ Ord B ) -> _E We A )
4	3	anim2i 593	. . 3 |- ( ( Tr A /\ ( A C_ B /\ Ord B ) ) -> ( Tr A /\ _E We A ) )
5	4	3impb 1260	. 2 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> ( Tr A /\ _E We A ) )
6		df-ord 5726	. 2 |- ( Ord A <-> ( Tr A /\ _E We A ) )
7	5, 6	sylibr 224	1 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   C_ wss 3574   Tr wtr 4752   _E cep 5028   We wwe 5072   Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-in 3581  df-ss 3588  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by:  ordin  5753  ssorduni  6985  suceloni  7013  ordom  7074  ordtypelem2  8424  hartogs  8449  card2on  8459  tskwe  8776  ondomon  9385  dford3lem2  37594  dford3  37595  iunord  42422