MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oneli Structured version   Unicode version

Theorem oneli 5540
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
StepHypRef Expression
1 on.1 . 2  |-  A  e.  On
2 onelon 5458 . 2  |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )
31, 2mpan 674 1  |-  ( B  e.  A  ->  B  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1867   Oncon0 5433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-ord 5436  df-on 5437
This theorem is referenced by:  onssneli  5542  oawordeulem  7254  rankuni  8324  tcrank  8345  cardne  8389  cardval2  8415  alephsuc2  8500  cfsmolem  8689  cfcof  8693  alephreg  8996  pwcfsdom  8997  tskcard  9195  onsucconi  30923
  Copyright terms: Public domain W3C validator