Theorem elon2 6368

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 3484	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		elong 6365	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	biadanii 833	. 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4	3	biancomi 467	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  Vcvv 3463  Ord word 6356  Oncon0 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-v 3465  df-ss 3930  df-uni 4874  df-tr 5220  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6360  df-on 6361

This theorem is referenced by:  ordsuci  7803  onsucb  7809  tfrlem12  8372  tfrlem13  8373  gruina  10799  bdayimaon  27819  noeta2  27916  etaslts2  27949  oldlim  28042  bdayons  28431  oaltublim  43902  omord2lim  43912  oaun3lem3  43988  nadd2rabon  43999  nadd1rabon  44003