Theorem elon2 6352

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 3474	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		elong 6349	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	biadanii 831	. 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4	3	biancomi 466	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  Vcvv 3453  Ord word 6340  Oncon0 6341

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 3919  df-uni 4863  df-tr 5205  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6344  df-on 6345

This theorem is referenced by:  ordsuci  7786  onsucb  7792  tfrlem12  8354  tfrlem13  8355  gruina  10770  bdayimaon  27745  noeta2  27842  etaslts2  27875  oldlim  27968  bdayons  28357  oaltublim  43828  omord2lim  43838  oaun3lem3  43914  nadd2rabon  43925  nadd1rabon  43929