Theorem onelon 4529

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 4520	. 2 |- ( A e. On -> Ord A )
2		ordelon 4528	. 2 |- ( ( Ord A /\ B e. A ) -> B e. On )
3	1, 2	sylan 283	1 |- ( ( A e. On /\ B e. A ) -> B e. On )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   Ord word 4507   Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513

This theorem is used by:  oneli  4573  ssorduni  4634  unon  4658  tfrlemibacc  6597  tfrlemibxssdm  6598  tfrlemibfn  6599  tfrexlem  6605  tfr1onlemsucaccv  6612  tfrcllemsucaccv  6625  sucinc2  6719  oav2  6736  omv2  6738