Theorem elong 6340

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 6339	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6336	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3647	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  Ord word 6331  Oncon0 6332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3449  df-ss 3931  df-uni 4872  df-tr 5215  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336

This theorem is referenced by:  elon  6341  eloni  6342  elon2  6343  ordelon  6356  onin  6363  limelon  6397  ordsssuc2  6425  onprc  7754  ssonuni  7756  sucexeloni  7785  sucexeloniOLD  7786  ordsucOLD  7789  cofon1  8636  cofon2  8637  enp1i  9224  oion  9489  hartogs  9497  card2on  9507  tskwe  9903  onssnum  9993  hsmexlem1  10379  ondomon  10516  1stcrestlem  23339  nosupno  27615  noinfno  27630  hfninf  36174  rn1st  45267