Theorem elon 6364

Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Hypothesis

Ref	Expression
elon.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elon	⊢ (𝐴 ∈ On ↔ Ord 𝐴)

Proof of Theorem elon

Step	Hyp	Ref	Expression
1		elon.1	. 2 ⊢ 𝐴 ∈ V
2		elong 6363	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2098  Vcvv 3466  Ord word 6354  Oncon0 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-v 3468  df-in 3948  df-ss 3958  df-uni 4901  df-tr 5257  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359

This theorem is referenced by:  tron  6378  0elon  6409  smogt  8363  dfrecs3  8368  dfrecs3OLD  8369  rdglim2  8428  omeulem1  8578  naddcllem  8672  isfinite2  9298  r0weon  10004  cflim3  10254  inar1  10767  addsproplem7  27811  ellimits  35378  dford3lem2  42280