Theorem onelon 4487

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 4478	. 2 |- ( A e. On -> Ord A )
2		ordelon 4486	. 2 |- ( ( Ord A /\ B e. A ) -> B e. On )
3	1, 2	sylan 283	1 |- ( ( A e. On /\ B e. A ) -> B e. On )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   Ord word 4465   Oncon0 4466

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471

This theorem is referenced by:  oneli  4531  ssorduni  4591  unon  4615  tfrlemibacc  6535  tfrlemibxssdm  6536  tfrlemibfn  6537  tfrexlem  6543  tfr1onlemsucaccv  6550  tfrcllemsucaccv  6563  sucinc2  6657  oav2  6674  omv2  6676