Theorem elon2 6372

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 3472	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		elong 6369	. . 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 2145  Vcvv 3451  Ord word 6360  Oncon0 6361

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-v 3453  df-ss 3916  df-uni 4868  df-tr 5213  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6364  df-on 6365

This theorem is used by:  ordsuci  7820  onsucb  7826  tfrlem12  8390  tfrlem13  8391  gruina  10896  bdayimaon  28043  noeta2  28140  etaslts2  28173  oldlim  28266  bdayons  28655  oaltublim  44276  omord2lim  44286  oaun3lem3  44362  nadd2rabon  44373  nadd1rabon  44377