Theorem elon2 6202

Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)

Assertion

Ref	Expression
elon2	⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Proof of Theorem elon2

Step	Hyp	Ref	Expression
1		elex 3416	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		elong 6199	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	biadanii 822	. 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4	3	biancomi 466	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  Vcvv 3398  Ord word 6190  Oncon0 6191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-v 3400  df-in 3860  df-ss 3870  df-uni 4806  df-tr 5147  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-ord 6194  df-on 6195

This theorem is referenced by:  sucelon  7574  tfrlem12  8103  tfrlem13  8104  gruina  10397  bdayimaon  33582  noeta2  33665  etasslt2  33694  oldlim  33755