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Theorem elong 5892
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 5891 . 2 (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2 df-on 5888 . 2 On = {𝑥 ∣ Ord 𝑥}
31, 2elab2g 3493 1 (𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2139  Ord word 5883  Oncon0 5884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-tr 4905  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888
This theorem is referenced by:  elon  5893  eloni  5894  elon2  5895  ordelon  5908  onin  5915  limelon  5949  ordsssuc2  5975  onprc  7150  ssonuni  7152  suceloni  7179  ordsuc  7180  oion  8608  hartogs  8616  card2on  8626  tskwe  8986  onssnum  9073  hsmexlem1  9460  ondomon  9597  1stcrestlem  21477  nosupno  32176  hfninf  32620
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