Theorem oneli 4985

Description: A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
oneli	|- ( B e. A -> B e. On )

Proof of Theorem oneli

Step	Hyp	Ref	Expression
1		on.1	. 2 |- A e. On
2		onelon 4903	. 2 |- ( ( A e. On /\ B e. A ) -> B e. On )
3	1, 2	mpan 670	1 |- ( B e. A -> B e. On )

Syntax hints:   -> wi 4   e. wcel 1767   Oncon0 4878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882

This theorem is referenced by:  onssneli  4987  oawordeulem  7204  rankuni  8282  tcrank  8303  cardne  8347  cardval2  8373  alephsuc2  8462  cfsmolem  8651  cfcof  8655  alephreg  8958  pwcfsdom  8959  tskcard  9160  onsucconi  29755