Theorem elon 4462

Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Hypothesis

Ref	Expression
elon.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elon	⊢ (𝐴 ∈ On ↔ Ord 𝐴)

Proof of Theorem elon

Step	Hyp	Ref	Expression
1		elon.1	. 2 ⊢ 𝐴 ∈ V
2		elong 4461	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)

Syntax hints:   ↔ wb 105   ∈ wcel 2200  Vcvv 2799  Ord word 4450  Oncon0 4451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888  df-tr 4182  df-iord 4454  df-on 4456

This theorem is referenced by:  tron  4470  0elon  4480  ordtriexmidlem  4608  ontr2exmid  4614  ordtri2or2exmidlem  4615  onsucelsucexmidlem  4618  exmidonfinlem  7359  bj-omelon  16254