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Metamath Proof Explorer
Description: If there is an untangled element of a class, then the intersection of
the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011)
|
|
Ref |
Expression |
|
Assertion |
untint |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀ 𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
intss1 |
⊢ ( 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥 ) |
| 2 |
|
ssralv |
⊢ ( ∩ 𝐴 ⊆ 𝑥 → ( ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀ 𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦 ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝑥 ∈ 𝐴 → ( ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀ 𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦 ) ) |
| 4 |
3
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀ 𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦 ) |