Theorem elon2 4354

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 4353	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		elex 2737	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
3	1, 2	jca 304	. 2 ⊢ (𝐴 ∈ On → (Ord 𝐴 ∧ 𝐴 ∈ V))
4		elong 4351	. . 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 2136  Vcvv 2726  Ord word 4340  Oncon0 4341

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  tfrexlem  6302  pw1on  7182