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Metamath Proof Explorer
Description: An ordinal number is an ordinal set. Part of Definition 1.2 of
Schloeder p. 1. (Contributed by NM, 8-Feb-2004)
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|
Ref |
Expression |
|
Assertion |
elon2 |
⊢ ( 𝐴 ∈ On ↔ ( Ord 𝐴 ∧ 𝐴 ∈ V ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
elex |
⊢ ( 𝐴 ∈ On → 𝐴 ∈ V ) |
| 2 |
|
elong |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴 ) ) |
| 3 |
1 2
|
biadanii |
⊢ ( 𝐴 ∈ On ↔ ( 𝐴 ∈ V ∧ Ord 𝐴 ) ) |
| 4 |
3
|
biancomi |
⊢ ( 𝐴 ∈ On ↔ ( Ord 𝐴 ∧ 𝐴 ∈ V ) ) |