Theorem elon2 4298

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 4297	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		elex 2697	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
3	1, 2	jca 304	. 2 ⊢ (𝐴 ∈ On → (Ord 𝐴 ∧ 𝐴 ∈ V))
4		elong 4295	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
5	4	biimparc 297	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ On)
6	3, 5	impbii 125	1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  Vcvv 2686  Ord word 4284  Oncon0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290

This theorem is referenced by:  tfrexlem  6231