Theorem elon2 6373

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 3476	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		elong 6370	. . 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 2143  Vcvv 3455  Ord word 6361  Oncon0 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-ss 3923  df-uni 4874  df-tr 5220  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366

This theorem is referenced by:  ordsuci  7808  onsucb  7814  tfrlem12  8377  tfrlem13  8378  gruina  10804  bdayimaon  27838  noeta2  27935  etaslts2  27968  oldlim  28061  bdayons  28450  oaltublim  44000  omord2lim  44010  oaun3lem3  44086  nadd2rabon  44097  nadd1rabon  44101