Theorem ontr1 4225

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 4211	. 2 |- ( C e. On -> Ord C )
2		ordtr1 4224	. 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 103   e. wcel 1439   Ord word 4198   Oncon0 4199

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204

This theorem is referenced by:  smoiun  6080  onunsnss  6681  snon0  6699  ltsopi  6940  prarloclemarch2  7039