Theorem onelon 4314

Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)

Assertion

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

Proof of Theorem onelon

Step	Hyp	Ref	Expression
1		eloni 4305	. 2 |- ( A e. On -> Ord A )
2		ordelon 4313	. 2 |- ( ( Ord A /\ B e. A ) -> B e. On )
3	1, 2	sylan 281	1 |- ( ( A e. On /\ B e. A ) -> B e. On )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1481   Ord word 4292   Oncon0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298

This theorem is referenced by:  oneli  4358  ssorduni  4411  unon  4435  tfrlemibacc  6231  tfrlemibxssdm  6232  tfrlemibfn  6233  tfrexlem  6239  tfr1onlemsucaccv  6246  tfrcllemsucaccv  6259  sucinc2  6350  oav2  6367  omv2  6369