Theorem ontr1 4486

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 4472	. 2 |- ( C e. On -> Ord C )
2		ordtr1 4485	. 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 2202   Ord word 4459   Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  smoiun  6466  nntr2  6670  onunsnss  7108  snon0  7133  exmidontriimlem2  7436  ltsopi  7539  prarloclemarch2  7638  pwle2  16599