Theorem elon2 4394

Description: An ordinal number is an ordinal set. (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		eloni 4393	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		elex 2763	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
3	1, 2	jca 306	. 2 ⊢ (𝐴 ∈ On → (Ord 𝐴 ∧ 𝐴 ∈ V))
4		elong 4391	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
5	4	biimparc 299	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ On)
6	3, 5	impbii 126	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2160  Vcvv 2752  Ord word 4380  Oncon0 4381

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386

This theorem is referenced by:  tfrexlem  6358  pw1on  7254