MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elong Structured version   Visualization version   GIF version

Theorem elong 5690
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 5689 . 2 (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2 df-on 5686 . 2 On = {𝑥 ∣ Ord 𝑥}
31, 2elab2g 3336 1 (𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 1987  Ord word 5681  Oncon0 5682
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 3188  df-in 3562  df-ss 3569  df-uni 4403  df-tr 4713  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686
This theorem is referenced by:  elon  5691  eloni  5692  elon2  5693  ordelon  5706  onin  5713  limelon  5747  ordsssuc2  5773  onprc  6931  ssonuni  6933  suceloni  6960  ordsuc  6961  oion  8385  hartogs  8393  card2on  8403  tskwe  8720  onssnum  8807  hsmexlem1  9192  ondomon  9329  1stcrestlem  21165  nosino  31575  hfninf  31935
  Copyright terms: Public domain W3C validator