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Theorem elon 5696
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Hypothesis
Ref Expression
elon.1 𝐴 ∈ V
Assertion
Ref Expression
elon (𝐴 ∈ On ↔ Ord 𝐴)

Proof of Theorem elon
StepHypRef Expression
1 elon.1 . 2 𝐴 ∈ V
2 elong 5695 . 2 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2ax-mp 5 1 (𝐴 ∈ On ↔ Ord 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1987  Vcvv 3189  Ord word 5686  Oncon0 5687
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 2912  df-rex 2913  df-v 3191  df-in 3566  df-ss 3573  df-uni 4408  df-tr 4718  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691
This theorem is referenced by:  tron  5710  0elon  5742  smogt  7416  dfrecs3  7421  rdglim2  7480  omeulem1  7614  isfinite2  8170  r0weon  8787  cflim3  9036  inar1  9549  ellimits  31694  dford3lem2  37109
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