Theorem elong 6259

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 6258	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6255	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3604	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2108  Ord word 6250  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-in 3890  df-ss 3900  df-uni 4837  df-tr 5188  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by:  elon  6260  eloni  6261  elon2  6262  ordelon  6275  onin  6282  limelon  6314  ordsssuc2  6339  onprc  7605  ssonuni  7607  suceloni  7635  ordsuc  7636  oion  9225  hartogs  9233  card2on  9243  tskwe  9639  onssnum  9727  hsmexlem1  10113  ondomon  10250  1stcrestlem  22511  nosupno  33833  noinfno  33848  hfninf  34415