Description: Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023)
Ref | Expression | ||
---|---|---|---|
Hypothesis | disjxun0.1 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 = ∅ ) | |
Assertion | disjxun0 | ⊢ ( 𝜑 → ( Disj 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjxun0.1 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 = ∅ ) | |
2 | nel02 | ⊢ ( 𝐶 = ∅ → ¬ 𝑦 ∈ 𝐶 ) | |
3 | 1 2 | syl | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ¬ 𝑦 ∈ 𝐶 ) |
4 | 3 | rmounid | ⊢ ( 𝜑 → ( ∃* 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝑦 ∈ 𝐶 ↔ ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) ) |
5 | 4 | albidv | ⊢ ( 𝜑 → ( ∀ 𝑦 ∃* 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝑦 ∈ 𝐶 ↔ ∀ 𝑦 ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) ) |
6 | df-disj | ⊢ ( Disj 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝐶 ↔ ∀ 𝑦 ∃* 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝑦 ∈ 𝐶 ) | |
7 | df-disj | ⊢ ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀ 𝑦 ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) | |
8 | 5 6 7 | 3bitr4g | ⊢ ( 𝜑 → ( Disj 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶 ) ) |