Theorem elong 6314

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 6313	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6310	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3631	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2111  Ord word 6305  Oncon0 6306

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-ss 3914  df-uni 4857  df-tr 5197  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309  df-on 6310

This theorem is referenced by:  elon  6315  eloni  6316  elon2  6317  ordelon  6330  onin  6337  limelon  6371  ordsssuc2  6399  onprc  7711  ssonuni  7713  sucexeloni  7742  cofon1  8587  cofon2  8588  enp1i  9163  oion  9422  hartogs  9430  card2on  9440  tskwe  9843  onssnum  9931  hsmexlem1  10317  ondomon  10454  1stcrestlem  23367  nosupno  27642  noinfno  27657  hfninf  36230  rn1st  45369