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 3471	. . 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 2144  Vcvv 3450  Ord word 6363  Oncon0 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3897  df-uni 4867  df-tr 5212  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6367  df-on 6368

This theorem is used by:  ordsuci  7824  onsucb  7830  tfrlem12  8402  tfrlem13  8403  gruina  10908  bdayimaon  28057  noeta2  28154  etaslts2  28187  oldlim  28280  bdayons  28669  oaltublim  44399  omord2lim  44409  oaun3lem3  44485  nadd2rabon  44496  nadd1rabon  44500