Theorem elong 6199

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 6198	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6195	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3578	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2112  Ord word 6190  Oncon0 6191

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 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-v 3400  df-in 3860  df-ss 3870  df-uni 4806  df-tr 5147  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-ord 6194  df-on 6195

This theorem is referenced by:  elon  6200  eloni  6201  elon2  6202  ordelon  6215  onin  6222  limelon  6254  ordsssuc2  6279  onprc  7540  ssonuni  7542  suceloni  7570  ordsuc  7571  oion  9130  hartogs  9138  card2on  9148  tskwe  9531  onssnum  9619  hsmexlem1  10005  ondomon  10142  1stcrestlem  22303  nosupno  33592  noinfno  33607  hfninf  34174