Theorem elon2 6375

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 3479	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		elong 6372	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	biadanii 834	. 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4	3	biancomi 468	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  Vcvv 3458  Ord word 6363  Oncon0 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-v 3460  df-ss 3925  df-uni 4876  df-tr 5222  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6367  df-on 6368

This theorem is used by:  ordsuci  7809  onsucb  7815  tfrlem12  8378  tfrlem13  8379  gruina  10813  bdayimaon  27872  noeta2  27969  etaslts2  28002  oldlim  28095  bdayons  28484  oaltublim  44049  omord2lim  44059  oaun3lem3  44135  nadd2rabon  44146  nadd1rabon  44150