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Theorem elon2 5703
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 3202 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 elong 5700 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2biadan2 673 . 2 (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4 ancom 466 . 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 384  wcel 1987  Vcvv 3190  Ord word 5691  Oncon0 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-in 3567  df-ss 3574  df-uni 4410  df-tr 4723  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696
This theorem is referenced by:  sucelon  6979  tfrlem12  7445  tfrlem13  7446  gruina  9600  nobndlem1  31608
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