Theorem ordtr1 4420

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 4410	. 2 |- ( Ord C -> Tr C )
2		trel 4135	. 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 2164   Tr wtr 4128   Ord word 4394

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167  df-uni 3837  df-tr 4129  df-iord 4398

This theorem is referenced by:  ontr1  4421  ordwe  4609  dfsmo2  6342  smores2  6349  smoel  6355  tfr1onlemsucaccv  6396  tfr1onlembxssdm  6398  tfr1onlembfn  6399  tfr1onlemaccex  6403  tfr1onlemres  6404  tfrcllemsucaccv  6409  tfrcllembxssdm  6411  tfrcllembfn  6412  tfrcllemaccex  6416  tfrcllemres  6417  tfrcl  6419  ordiso2  7096