Metamath Proof Explorer


Theorem sssymdifcl

Description: The class of all subsets of a class is closed under symmetric difference. (Contributed by RP, 3-Jan-2020)

Ref Expression
Hypothesis ssficl.a 𝐴 = { 𝑧𝑧𝐵 }
Assertion sssymdifcl 𝑥𝐴𝑦𝐴 ( ( 𝑥𝑦 ) ∪ ( 𝑦𝑥 ) ) ∈ 𝐴

Proof

Step Hyp Ref Expression
1 ssficl.a 𝐴 = { 𝑧𝑧𝐵 }
2 vex 𝑥 ∈ V
3 2 difexi ( 𝑥𝑦 ) ∈ V
4 vex 𝑦 ∈ V
5 4 difexi ( 𝑦𝑥 ) ∈ V
6 3 5 unex ( ( 𝑥𝑦 ) ∪ ( 𝑦𝑥 ) ) ∈ V
7 sseq1 ( 𝑧 = ( ( 𝑥𝑦 ) ∪ ( 𝑦𝑥 ) ) → ( 𝑧𝐵 ↔ ( ( 𝑥𝑦 ) ∪ ( 𝑦𝑥 ) ) ⊆ 𝐵 ) )
8 sseq1 ( 𝑧 = 𝑥 → ( 𝑧𝐵𝑥𝐵 ) )
9 sseq1 ( 𝑧 = 𝑦 → ( 𝑧𝐵𝑦𝐵 ) )
10 ssdifss ( 𝑥𝐵 → ( 𝑥𝑦 ) ⊆ 𝐵 )
11 ssdifss ( 𝑦𝐵 → ( 𝑦𝑥 ) ⊆ 𝐵 )
12 unss ( ( ( 𝑥𝑦 ) ⊆ 𝐵 ∧ ( 𝑦𝑥 ) ⊆ 𝐵 ) ↔ ( ( 𝑥𝑦 ) ∪ ( 𝑦𝑥 ) ) ⊆ 𝐵 )
13 12 biimpi ( ( ( 𝑥𝑦 ) ⊆ 𝐵 ∧ ( 𝑦𝑥 ) ⊆ 𝐵 ) → ( ( 𝑥𝑦 ) ∪ ( 𝑦𝑥 ) ) ⊆ 𝐵 )
14 10 11 13 syl2an ( ( 𝑥𝐵𝑦𝐵 ) → ( ( 𝑥𝑦 ) ∪ ( 𝑦𝑥 ) ) ⊆ 𝐵 )
15 1 6 7 8 9 14 cllem0 𝑥𝐴𝑦𝐴 ( ( 𝑥𝑦 ) ∪ ( 𝑦𝑥 ) ) ∈ 𝐴