Theorem ordtr1 4436

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 4426	. 2 |- ( Ord C -> Tr C )
2		trel 4150	. 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 2176   Tr wtr 4143   Ord word 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4144  df-iord 4414

This theorem is referenced by:  ontr1  4437  ordwe  4625  dfsmo2  6375  smores2  6382  smoel  6388  tfr1onlemsucaccv  6429  tfr1onlembxssdm  6431  tfr1onlembfn  6432  tfr1onlemaccex  6436  tfr1onlemres  6437  tfrcllemsucaccv  6442  tfrcllembxssdm  6444  tfrcllembfn  6445  tfrcllemaccex  6449  tfrcllemres  6450  tfrcl  6452  ordiso2  7139