Theorem ordtr1 4479

Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)

Assertion

Ref	Expression
ordtr1	|- ( Ord C -> ( ( A e. B /\ B e. C ) -> A e. C ) )

Proof of Theorem ordtr1

Step	Hyp	Ref	Expression
1		ordtr 4469	. 2 |- ( Ord C -> Tr C )
2		trel 4189	. 2 |- ( Tr C -> ( ( A e. B /\ B e. C ) -> A e. C ) )
3	1, 2	syl 14	1 |- ( Ord C -> ( ( A e. B /\ B e. C ) -> A e. C ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   Tr wtr 4182   Ord word 4453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-uni 3889  df-tr 4183  df-iord 4457

This theorem is referenced by:  ontr1  4480  ordwe  4668  dfsmo2  6433  smores2  6440  smoel  6446  tfr1onlemsucaccv  6487  tfr1onlembxssdm  6489  tfr1onlembfn  6490  tfr1onlemaccex  6494  tfr1onlemres  6495  tfrcllemsucaccv  6500  tfrcllembxssdm  6502  tfrcllembfn  6503  tfrcllemaccex  6507  tfrcllemres  6508  tfrcl  6510  ordiso2  7202