Theorem elong 6385

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 6384	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6381	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3637	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2147  Ord word 6376  Oncon0 6377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-uni 4871  df-tr 5216  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6380  df-on 6381

This theorem is used by:  elon  6386  eloni  6387  elon2  6388  ordelon  6401  onin  6409  limelon  6443  ordsssuc2  6471  onprc  7813  ssonuni  7815  sucexeloni  7844  cofon1  8706  cofon2  8707  enp1i  9295  oion  9554  hartogs  9562  card2on  9572  tskwe  10055  onssnum  10143  hsmexlem1  10528  ondomon  10671  1stcrestlem  23797  nosupno  28087  noinfno  28102  hfninf  37112  rn1st  46488