Theorem elon 6351

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 6350	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)

Syntax hints:   ↔ wb 208   ∈ wcel 2141  Vcvv 3453  Ord word 6341  Oncon0 6342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-v 3455  df-ss 3921  df-uni 4865  df-tr 5207  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-ord 6345  df-on 6346

This theorem is referenced by:  tron  6365  0elon  6397  smogt  8333  dfrecs3  8338  rdglim2  8398  omeulem1  8546  naddcllem  8641  isfinite2  9238  r0weon  9965  cflim3  10216  inar1  10730  addsproplem7  28045  ellimits  36222  dford3lem2  43568