MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onordi Structured version   Visualization version   GIF version

Theorem onordi 5975
Description: An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
onordi Ord 𝐴

Proof of Theorem onordi
StepHypRef Expression
1 on.1 . 2 𝐴 ∈ On
2 eloni 5876 . 2 (𝐴 ∈ On → Ord 𝐴)
31, 2ax-mp 5 1 Ord 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2144  Ord word 5865  Oncon0 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-in 3728  df-ss 3735  df-uni 4573  df-tr 4885  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870
This theorem is referenced by:  ontrci  5976  onirri  5977  onun2i  5986  onuniorsuci  7185  onsucssi  7187  oawordeulem  7787  omopthi  7890  bndrank  8867  rankprb  8877  rankuniss  8892  rankelun  8898  rankelpr  8899  rankelop  8900  rankmapu  8904  rankxplim3  8907  rankxpsuc  8908  cardlim  8997  carduni  9006  dfac8b  9053  alephdom2  9109  alephfp  9130  dfac12lem2  9167  pm110.643ALT  9201  cfsmolem  9293  ttukeylem6  9537  ttukeylem7  9538  unsnen  9576  efgmnvl  18333  slerec  32254  hfuni  32622  finxpsuclem  33564  pwfi2f1o  38185
  Copyright terms: Public domain W3C validator