Theorem elong 6372

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 6371	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6368	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3647	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2150  Ord word 6363  Oncon0 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-v 3464  df-ss 3930  df-uni 4878  df-tr 5224  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-ord 6367  df-on 6368

This theorem is referenced by:  elon  6373  eloni  6374  elon2  6375  ordelon  6388  onin  6396  limelon  6430  ordsssuc2  6458  onprc  7780  ssonuni  7782  sucexeloni  7811  cofon1  8661  cofon2  8662  enp1i  9242  oion  9501  hartogs  9509  card2on  9519  tskwe  9939  onssnum  10027  hsmexlem1  10413  ondomon  10550  1stcrestlem  23592  nosupno  27847  noinfno  27862  hfninf  36636  rn1st  45940