Description: Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | eliinid | ⊢ ( ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ∈ 𝐶 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl | ⊢ ( ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ) | |
2 | eliin | ⊢ ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) ) | |
3 | 2 | adantr | ⊢ ( ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) ) |
4 | 1 3 | mpbid | ⊢ ( ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵 ) → ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) |
5 | rspa | ⊢ ( ( ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ∈ 𝐶 ) | |
6 | 4 5 | sylancom | ⊢ ( ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ∈ 𝐶 ) |