Theorem elong 6335

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 6334	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6331	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3637	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Ord word 6326  Oncon0 6327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-ss 3920  df-uni 4866  df-tr 5208  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-ord 6330  df-on 6331

This theorem is referenced by:  elon  6336  eloni  6337  elon2  6338  ordelon  6351  onin  6358  limelon  6392  ordsssuc2  6420  onprc  7735  ssonuni  7737  sucexeloni  7766  cofon1  8612  cofon2  8613  enp1i  9193  oion  9455  hartogs  9463  card2on  9473  tskwe  9876  onssnum  9964  hsmexlem1  10350  ondomon  10487  1stcrestlem  23413  nosupno  27688  noinfno  27703  hfninf  36408  rn1st  45660