Theorem elong 6328

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 6327	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6324	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3644	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  Ord word 6319  Oncon0 6320

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 3446  df-ss 3928  df-uni 4868  df-tr 5210  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324

This theorem is referenced by:  elon  6329  eloni  6330  elon2  6331  ordelon  6344  onin  6351  limelon  6385  ordsssuc2  6413  onprc  7734  ssonuni  7736  sucexeloni  7765  sucexeloniOLD  7766  ordsucOLD  7769  cofon1  8613  cofon2  8614  enp1i  9200  oion  9465  hartogs  9473  card2on  9483  tskwe  9879  onssnum  9969  hsmexlem1  10355  ondomon  10492  1stcrestlem  23372  nosupno  27648  noinfno  27663  hfninf  36167  rn1st  45260