Step |
Hyp |
Ref |
Expression |
1 |
|
ssficl.a |
⊢ 𝐴 = { 𝑧 ∣ 𝑧 ⊆ 𝐵 } |
2 |
|
vex |
⊢ 𝑥 ∈ V |
3 |
2
|
difexi |
⊢ ( 𝑥 ∖ 𝑦 ) ∈ V |
4 |
|
vex |
⊢ 𝑦 ∈ V |
5 |
4
|
difexi |
⊢ ( 𝑦 ∖ 𝑥 ) ∈ V |
6 |
3 5
|
unex |
⊢ ( ( 𝑥 ∖ 𝑦 ) ∪ ( 𝑦 ∖ 𝑥 ) ) ∈ V |
7 |
|
sseq1 |
⊢ ( 𝑧 = ( ( 𝑥 ∖ 𝑦 ) ∪ ( 𝑦 ∖ 𝑥 ) ) → ( 𝑧 ⊆ 𝐵 ↔ ( ( 𝑥 ∖ 𝑦 ) ∪ ( 𝑦 ∖ 𝑥 ) ) ⊆ 𝐵 ) ) |
8 |
|
sseq1 |
⊢ ( 𝑧 = 𝑥 → ( 𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵 ) ) |
9 |
|
sseq1 |
⊢ ( 𝑧 = 𝑦 → ( 𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵 ) ) |
10 |
|
ssdifss |
⊢ ( 𝑥 ⊆ 𝐵 → ( 𝑥 ∖ 𝑦 ) ⊆ 𝐵 ) |
11 |
|
ssdifss |
⊢ ( 𝑦 ⊆ 𝐵 → ( 𝑦 ∖ 𝑥 ) ⊆ 𝐵 ) |
12 |
|
unss |
⊢ ( ( ( 𝑥 ∖ 𝑦 ) ⊆ 𝐵 ∧ ( 𝑦 ∖ 𝑥 ) ⊆ 𝐵 ) ↔ ( ( 𝑥 ∖ 𝑦 ) ∪ ( 𝑦 ∖ 𝑥 ) ) ⊆ 𝐵 ) |
13 |
12
|
biimpi |
⊢ ( ( ( 𝑥 ∖ 𝑦 ) ⊆ 𝐵 ∧ ( 𝑦 ∖ 𝑥 ) ⊆ 𝐵 ) → ( ( 𝑥 ∖ 𝑦 ) ∪ ( 𝑦 ∖ 𝑥 ) ) ⊆ 𝐵 ) |
14 |
10 11 13
|
syl2an |
⊢ ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵 ) → ( ( 𝑥 ∖ 𝑦 ) ∪ ( 𝑦 ∖ 𝑥 ) ) ⊆ 𝐵 ) |
15 |
1 6 7 8 9 14
|
cllem0 |
⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝑥 ∖ 𝑦 ) ∪ ( 𝑦 ∖ 𝑥 ) ) ∈ 𝐴 |