MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elon2 Structured version   Visualization version   GIF version

Theorem elon2 6397
Description: An ordinal number is an ordinal set. Part of Definition 1.2 of [Schloeder] p. 1. (Contributed by NM, 8-Feb-2004.)
Assertion
Ref Expression
elon2 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))

Proof of Theorem elon2
StepHypRef Expression
1 elex 3499 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 elong 6394 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2biadanii 822 . 2 (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
43biancomi 462 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  Vcvv 3478  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-uni 4913  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  ordsuci  7828  onsucb  7837  tfrlem12  8428  tfrlem13  8429  gruina  10856  bdayimaon  27753  noeta2  27844  etasslt2  27874  oldlim  27940  oaltublim  43280  omord2lim  43290  oaun3lem3  43366  nadd2rabon  43377  nadd1rabon  43381
  Copyright terms: Public domain W3C validator