Theorem elon 4471

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 4470	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)

Syntax hints:   ↔ wb 105   ∈ wcel 2202  Vcvv 2802  Ord word 4459  Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  tron  4479  0elon  4489  ordtriexmidlem  4617  ontr2exmid  4623  ordtri2or2exmidlem  4624  onsucelsucexmidlem  4627  exmidonfinlem  7404  bj-omelon  16582