Theorem ontr1 4391

Description: Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)

Assertion

Ref	Expression
ontr1	|- ( C e. On -> ( ( A e. B /\ B e. C ) -> A e. C ) )

Proof of Theorem ontr1

Step	Hyp	Ref	Expression
1		eloni 4377	. 2 |- ( C e. On -> Ord C )
2		ordtr1 4390	. 2 |- ( Ord C -> ( ( A e. B /\ B e. C ) -> A e. C ) )
3	1, 2	syl 14	1 |- ( C e. On -> ( ( A e. B /\ B e. C ) -> A e. C ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   Ord word 4364   Oncon0 4365

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370

This theorem is referenced by:  smoiun  6304  nntr2  6506  onunsnss  6918  snon0  6937  exmidontriimlem2  7223  ltsopi  7321  prarloclemarch2  7420  pwle2  14833