Theorem ordtr1 4453

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 4443	. 2 |- ( Ord C -> Tr C )
2		trel 4165	. 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 2178   Tr wtr 4158   Ord word 4427

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431

This theorem is referenced by:  ontr1  4454  ordwe  4642  dfsmo2  6396  smores2  6403  smoel  6409  tfr1onlemsucaccv  6450  tfr1onlembxssdm  6452  tfr1onlembfn  6453  tfr1onlemaccex  6457  tfr1onlemres  6458  tfrcllemsucaccv  6463  tfrcllembxssdm  6465  tfrcllembfn  6466  tfrcllemaccex  6470  tfrcllemres  6471  tfrcl  6473  ordiso2  7163