Theorem elon2 6389

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 6386	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	biadanii 820	. 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4	3	biancomi 461	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∈ wcel 2100  Vcvv 3472  Ord word 6377  Oncon0 6378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-v 3474  df-ss 3975  df-uni 4917  df-tr 5272  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-ord 6381  df-on 6382

This theorem is referenced by:  ordsuci  7822  onsucb  7831  tfrlem12  8423  tfrlem13  8424  gruina  10862  bdayimaon  27720  noeta2  27811  etasslt2  27841  oldlim  27907  oaltublim  43068  omord2lim  43078  oaun3lem3  43154  nadd2rabon  43165  nadd1rabon  43169