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

Theorem elon2 6395
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 3501 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 elong 6392 . . 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 2108  Vcvv 3480  Ord word 6383  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  ordsuci  7828  onsucb  7837  tfrlem12  8429  tfrlem13  8430  gruina  10858  bdayimaon  27738  noeta2  27829  etasslt2  27859  oldlim  27925  oaltublim  43303  omord2lim  43313  oaun3lem3  43389  nadd2rabon  43400  nadd1rabon  43404
  Copyright terms: Public domain W3C validator