Theorem elong 6393

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 6392	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6389	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3682	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2105  Ord word 6384  Oncon0 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-v 3479  df-ss 3979  df-uni 4912  df-tr 5265  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389

This theorem is referenced by:  elon  6394  eloni  6395  elon2  6396  ordelon  6409  onin  6416  limelon  6449  ordsssuc2  6476  onprc  7796  ssonuni  7798  sucexeloni  7828  sucexeloniOLD  7829  suceloniOLD  7831  ordsucOLD  7833  cofon1  8708  cofon2  8709  enp1i  9310  oion  9573  hartogs  9581  card2on  9591  tskwe  9987  onssnum  10077  hsmexlem1  10463  ondomon  10600  1stcrestlem  23475  nosupno  27762  noinfno  27777  hfninf  36167  rn1st  45218