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Theorem elon2 5772
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 3243 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 elong 5769 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2biadan2 675 . 2 (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4 ancom 465 . 2 ((𝐴 ∈ V ∧ Ord 𝐴) ↔ (Ord 𝐴𝐴 ∈ V))
53, 4bitri 264 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030  Vcvv 3231  Ord word 5760  Oncon0 5761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-uni 4469  df-tr 4786  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765
This theorem is referenced by:  sucelon  7059  tfrlem12  7530  tfrlem13  7531  gruina  9678  bdayimaon  31968
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