Theorem oneli 4937

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 4855	. 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 1758   Oncon0 4830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834

This theorem is referenced by:  onssneli  4939  oawordeulem  7106  rankuni  8184  tcrank  8205  cardne  8249  cardval2  8275  alephsuc2  8364  cfsmolem  8553  cfcof  8557  alephreg  8860  pwcfsdom  8861  tskcard  9062  onsucconi  28447