Theorem ordtr1 4435

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 4425	. 2 |- ( Ord C -> Tr C )
2		trel 4149	. 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 4142   Ord word 4409

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 4143  df-iord 4413

This theorem is referenced by:  ontr1  4436  ordwe  4624  dfsmo2  6373  smores2  6380  smoel  6386  tfr1onlemsucaccv  6427  tfr1onlembxssdm  6429  tfr1onlembfn  6430  tfr1onlemaccex  6434  tfr1onlemres  6435  tfrcllemsucaccv  6440  tfrcllembxssdm  6442  tfrcllembfn  6443  tfrcllemaccex  6447  tfrcllemres  6448  tfrcl  6450  ordiso2  7137