Theorem elong 6322

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 6321	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6318	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3620	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2121  Ord word 6313  Oncon0 6314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-v 3435  df-ss 3902  df-uni 4842  df-tr 5183  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318

This theorem is referenced by:  elon  6323  eloni  6324  elon2  6325  ordelon  6338  onin  6345  limelon  6379  ordsssuc2  6407  onprc  7725  ssonuni  7727  sucexeloni  7756  cofon1  8602  cofon2  8603  enp1i  9183  oion  9445  hartogs  9453  card2on  9463  tskwe  9869  onssnum  9957  hsmexlem1  10343  ondomon  10480  1stcrestlem  23439  nosupno  27689  noinfno  27704  hfninf  36429  rn1st  45731