Theorem elong 6370

Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Assertion

Ref	Expression
elong	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Proof of Theorem elong

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordeq 6369	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6366	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3670	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2107  Ord word 6361  Oncon0 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3955  df-ss 3965  df-uni 4909  df-tr 5266  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6365  df-on 6366

This theorem is referenced by:  elon  6371  eloni  6372  elon2  6373  ordelon  6386  onin  6393  limelon  6426  ordsssuc2  6453  onprc  7762  ssonuni  7764  sucexeloni  7794  sucexeloniOLD  7795  suceloniOLD  7797  ordsucOLD  7799  cofon1  8668  cofon2  8669  enp1i  9276  oion  9528  hartogs  9536  card2on  9546  tskwe  9942  onssnum  10032  hsmexlem1  10418  ondomon  10555  1stcrestlem  22948  nosupno  27196  noinfno  27211  hfninf  35147  rn1st  43965