Description: The class of all supersets of a class is closed under binary union. (Contributed by RP, 3-Jan-2020)
Ref | Expression | ||
---|---|---|---|
Hypothesis | superficl.a | ⊢ 𝐴 = { 𝑧 ∣ 𝐵 ⊆ 𝑧 } | |
Assertion | superuncl | ⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∪ 𝑦 ) ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | superficl.a | ⊢ 𝐴 = { 𝑧 ∣ 𝐵 ⊆ 𝑧 } | |
2 | vex | ⊢ 𝑥 ∈ V | |
3 | vex | ⊢ 𝑦 ∈ V | |
4 | 2 3 | unex | ⊢ ( 𝑥 ∪ 𝑦 ) ∈ V |
5 | sseq2 | ⊢ ( 𝑧 = ( 𝑥 ∪ 𝑦 ) → ( 𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ ( 𝑥 ∪ 𝑦 ) ) ) | |
6 | sseq2 | ⊢ ( 𝑧 = 𝑥 → ( 𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥 ) ) | |
7 | sseq2 | ⊢ ( 𝑧 = 𝑦 → ( 𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦 ) ) | |
8 | ssun3 | ⊢ ( 𝐵 ⊆ 𝑥 → 𝐵 ⊆ ( 𝑥 ∪ 𝑦 ) ) | |
9 | 8 | adantr | ⊢ ( ( 𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦 ) → 𝐵 ⊆ ( 𝑥 ∪ 𝑦 ) ) |
10 | 1 4 5 6 7 9 | cllem0 | ⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∪ 𝑦 ) ∈ 𝐴 |