Theorem elong 6323

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 6322	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6319	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3633	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2113  Ord word 6314  Oncon0 6315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-v 3440  df-ss 3916  df-uni 4862  df-tr 5204  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319

This theorem is referenced by:  elon  6324  eloni  6325  elon2  6326  ordelon  6339  onin  6346  limelon  6380  ordsssuc2  6408  onprc  7721  ssonuni  7723  sucexeloni  7752  cofon1  8598  cofon2  8599  enp1i  9177  oion  9439  hartogs  9447  card2on  9457  tskwe  9860  onssnum  9948  hsmexlem1  10334  ondomon  10471  1stcrestlem  23394  nosupno  27669  noinfno  27684  hfninf  36329  rn1st  45459