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Theorem elong 6192
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 6191 . 2 (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2 df-on 6188 . 2 On = {𝑥 ∣ Ord 𝑥}
31, 2elab2g 3665 1 (𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wcel 2105  Ord word 6183  Oncon0 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-in 3940  df-ss 3949  df-uni 4831  df-tr 5164  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by:  elon  6193  eloni  6194  elon2  6195  ordelon  6208  onin  6215  limelon  6247  ordsssuc2  6272  onprc  7488  ssonuni  7490  suceloni  7517  ordsuc  7518  oion  8988  hartogs  8996  card2on  9006  tskwe  9367  onssnum  9454  hsmexlem1  9836  ondomon  9973  1stcrestlem  21988  nosupno  33100  hfninf  33544
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