Metamath Proof Explorer
Description: An alternate definition of class membership when the class is a set.
(Contributed by NM, 18-Aug-1993)
|
|
Ref |
Expression |
|
Hypothesis |
clel3.1 |
⊢ 𝐵 ∈ V |
|
Assertion |
clel3 |
⊢ ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
clel3.1 |
⊢ 𝐵 ∈ V |
2 |
|
clel3g |
⊢ ( 𝐵 ∈ V → ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥 ) ) ) |
3 |
1 2
|
ax-mp |
⊢ ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥 ) ) |