Metamath Proof Explorer
Description: A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011)
|
|
Ref |
Expression |
|
Assertion |
ifr0 |
⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
equid |
⊢ 𝑥 = 𝑥 |
2 |
|
vex |
⊢ 𝑥 ∈ V |
3 |
2
|
ideq |
⊢ ( 𝑥 I 𝑥 ↔ 𝑥 = 𝑥 ) |
4 |
1 3
|
mpbir |
⊢ 𝑥 I 𝑥 |
5 |
|
frirr |
⊢ ( ( I Fr 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝑥 I 𝑥 ) |
6 |
5
|
ex |
⊢ ( I Fr 𝐴 → ( 𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥 ) ) |
7 |
4 6
|
mt2i |
⊢ ( I Fr 𝐴 → ¬ 𝑥 ∈ 𝐴 ) |
8 |
7
|
eq0rdv |
⊢ ( I Fr 𝐴 → 𝐴 = ∅ ) |
9 |
|
fr0 |
⊢ I Fr ∅ |
10 |
|
freq2 |
⊢ ( 𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅ ) ) |
11 |
9 10
|
mpbiri |
⊢ ( 𝐴 = ∅ → I Fr 𝐴 ) |
12 |
8 11
|
impbii |
⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅ ) |