Description: The class of all subsets of a class is closed under class difference. (Contributed by RP, 3-Jan-2020)
Ref | Expression | ||
---|---|---|---|
Hypothesis | ssficl.a | ⊢ 𝐴 = { 𝑧 ∣ 𝑧 ⊆ 𝐵 } | |
Assertion | ssdifcl | ⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∖ 𝑦 ) ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssficl.a | ⊢ 𝐴 = { 𝑧 ∣ 𝑧 ⊆ 𝐵 } | |
2 | vex | ⊢ 𝑥 ∈ V | |
3 | 2 | difexi | ⊢ ( 𝑥 ∖ 𝑦 ) ∈ V |
4 | sseq1 | ⊢ ( 𝑧 = ( 𝑥 ∖ 𝑦 ) → ( 𝑧 ⊆ 𝐵 ↔ ( 𝑥 ∖ 𝑦 ) ⊆ 𝐵 ) ) | |
5 | sseq1 | ⊢ ( 𝑧 = 𝑥 → ( 𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵 ) ) | |
6 | sseq1 | ⊢ ( 𝑧 = 𝑦 → ( 𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵 ) ) | |
7 | ssdifss | ⊢ ( 𝑥 ⊆ 𝐵 → ( 𝑥 ∖ 𝑦 ) ⊆ 𝐵 ) | |
8 | 7 | adantr | ⊢ ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵 ) → ( 𝑥 ∖ 𝑦 ) ⊆ 𝐵 ) |
9 | 1 3 4 5 6 8 | cllem0 | ⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∖ 𝑦 ) ∈ 𝐴 |