Theorem elon 6370

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 6369	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2106  Vcvv 3474  Ord word 6360  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-v 3476  df-in 3954  df-ss 3964  df-uni 4908  df-tr 5265  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365

This theorem is referenced by:  tron  6384  0elon  6415  smogt  8363  dfrecs3  8368  dfrecs3OLD  8369  rdglim2  8428  omeulem1  8578  naddcllem  8671  isfinite2  9297  r0weon  10003  cflim3  10253  inar1  10766  addsproplem7  27448  ellimits  34870  dford3lem2  41751