Theorem elon2 6367

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 3493	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		elong 6364	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	biadanii 821	. 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4	3	biancomi 464	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  Vcvv 3475  Ord word 6355  Oncon0 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3953  df-ss 3963  df-uni 4905  df-tr 5262  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6359  df-on 6360

This theorem is referenced by:  ordsuci  7783  onsucb  7792  tfrlem12  8376  tfrlem13  8377  gruina  10800  bdayimaon  27163  noeta2  27253  etasslt2  27282  oldlim  27348  oaltublim  41911  omord2lim  41921  oaun3lem3  41997  nadd2rabon  42008  nadd1rabon  42012