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Theorem elon2 6406
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 3509 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 elong 6403 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2biadanii 821 . 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 3488  Ord word 6394  Oncon0 6395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490  df-ss 3993  df-uni 4932  df-tr 5284  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399
This theorem is referenced by:  ordsuci  7844  onsucb  7853  tfrlem12  8445  tfrlem13  8446  gruina  10887  bdayimaon  27756  noeta2  27847  etasslt2  27877  oldlim  27943  oaltublim  43252  omord2lim  43262  oaun3lem3  43338  nadd2rabon  43349  nadd1rabon  43353
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