Theorem elon2 6343

Description: An ordinal number is an ordinal set. Part of Definition 1.2 of [Schloeder] p. 1. (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 3468	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		elong 6340	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	biadanii 821	. 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4	3	biancomi 462	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3447  Ord word 6331  Oncon0 6332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3449  df-ss 3931  df-uni 4872  df-tr 5215  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336

This theorem is referenced by:  ordsuci  7784  onsucb  7792  tfrlem12  8357  tfrlem13  8358  gruina  10771  bdayimaon  27605  noeta2  27696  etasslt2  27726  oldlim  27798  bdayon  28173  oaltublim  43279  omord2lim  43289  oaun3lem3  43365  nadd2rabon  43376  nadd1rabon  43380