Theorem ordtr1 4419

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 4409	. 2 |- ( Ord C -> Tr C )
2		trel 4134	. 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 4127   Ord word 4393

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 3159  df-ss 3166  df-uni 3836  df-tr 4128  df-iord 4397

This theorem is referenced by:  ontr1  4420  ordwe  4608  dfsmo2  6340  smores2  6347  smoel  6353  tfr1onlemsucaccv  6394  tfr1onlembxssdm  6396  tfr1onlembfn  6397  tfr1onlemaccex  6401  tfr1onlemres  6402  tfrcllemsucaccv  6407  tfrcllembxssdm  6409  tfrcllembfn  6410  tfrcllemaccex  6414  tfrcllemres  6415  tfrcl  6417  ordiso2  7094