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Metamath Proof Explorer
Description: Equinumerosity is reflexive. Theorem 1 of Suppes p. 92. (Contributed by NM, 18-Jun-1998) (Revised by Mario Carneiro, 26-Apr-2015)
|
|
Ref |
Expression |
|
Assertion |
enrefg |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
f1oi |
⊢ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 |
| 2 |
|
f1oen2g |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 ) → 𝐴 ≈ 𝐴 ) |
| 3 |
1 2
|
mp3an3 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) → 𝐴 ≈ 𝐴 ) |
| 4 |
3
|
anidms |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴 ) |