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Theorem elon2 6174
Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
Assertion
Ref Expression
elon2 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))

Proof of Theorem elon2
StepHypRef Expression
1 elex 3462 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 elong 6171 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2biadanii 821 . 2 (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
43biancomi 466 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2112  Vcvv 3444  Ord word 6162  Oncon0 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-in 3891  df-ss 3901  df-uni 4804  df-tr 5140  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167
This theorem is referenced by:  sucelon  7516  tfrlem12  8012  tfrlem13  8013  gruina  10233  bdayimaon  33311
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