Theorem elong 6326

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 6325	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6322	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3636	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Ord word 6317  Oncon0 6318

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 3443  df-ss 3919  df-uni 4865  df-tr 5207  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322

This theorem is referenced by:  elon  6327  eloni  6328  elon2  6329  ordelon  6342  onin  6349  limelon  6383  ordsssuc2  6411  onprc  7725  ssonuni  7727  sucexeloni  7756  cofon1  8602  cofon2  8603  enp1i  9183  oion  9445  hartogs  9453  card2on  9463  tskwe  9866  onssnum  9954  hsmexlem1  10340  ondomon  10477  1stcrestlem  23400  nosupno  27675  noinfno  27690  hfninf  36382  rn1st  45584