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Theorem elong 5719
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
elong (𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Proof of Theorem elong
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordeq 5718 . 2 (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2 df-on 5715 . 2 On = {𝑥 ∣ Ord 𝑥}
31, 2elab2g 3347 1 (𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 1988  Ord word 5710  Oncon0 5711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-in 3574  df-ss 3581  df-uni 4428  df-tr 4744  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715
This theorem is referenced by:  elon  5720  eloni  5721  elon2  5722  ordelon  5735  onin  5742  limelon  5776  ordsssuc2  5802  onprc  6969  ssonuni  6971  suceloni  6998  ordsuc  6999  oion  8426  hartogs  8434  card2on  8444  tskwe  8761  onssnum  8848  hsmexlem1  9233  ondomon  9370  1stcrestlem  21236  nosupno  31823  hfninf  32268
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