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

Theorem elong 5958
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 5957 . 2 (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2 df-on 5954 . 2 On = {𝑥 ∣ Ord 𝑥}
31, 2elab2g 3559 1 (𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wcel 2157  Ord word 5949  Oncon0 5950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-in 3787  df-ss 3794  df-uni 4642  df-tr 4958  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-ord 5953  df-on 5954
This theorem is referenced by:  elon  5959  eloni  5960  elon2  5961  ordelon  5974  onin  5981  limelon  6014  ordsssuc2  6039  onprc  7224  ssonuni  7226  suceloni  7253  ordsuc  7254  oion  8690  hartogs  8698  card2on  8708  tskwe  9069  onssnum  9156  hsmexlem1  9543  ondomon  9680  1stcrestlem  21490  nosupno  32192  hfninf  32636
  Copyright terms: Public domain W3C validator