Description: Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ixpssixp.1 | ⊢ Ⅎ 𝑥 𝜑 | |
ixpssixp.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 ) | ||
Assertion | ixpssixp | ⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 ⊆ X 𝑥 ∈ 𝐴 𝐶 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpssixp.1 | ⊢ Ⅎ 𝑥 𝜑 | |
2 | ixpssixp.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 ) | |
3 | 2 | ex | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶 ) ) |
4 | 1 3 | ralrimi | ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ) |
5 | ss2ixp | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X 𝑥 ∈ 𝐴 𝐵 ⊆ X 𝑥 ∈ 𝐴 𝐶 ) | |
6 | 4 5 | syl | ⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 ⊆ X 𝑥 ∈ 𝐴 𝐶 ) |