Theorem sssymdifcl 44716

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

Hypothesis

Ref	Expression
ssficl.a	⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵}

Assertion

Ref	Expression
sssymdifcl	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ 𝐴

Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴   𝑧,𝐵

Allowed substitution hints:   𝐴(𝑥, 𝑧)   𝐵(𝑥, 𝑦)

Proof of Theorem sssymdifcl

Step	Hyp	Ref	Expression
1		ssficl.a	. 2 ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵}
2		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
3	2	difexi 5296	. . 3 ⊢ (𝑥 ∖ 𝑦) ∈ V
4		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
5	4	difexi 5296	. . 3 ⊢ (𝑦 ∖ 𝑥) ∈ V
6	3, 5	unex 7771	. 2 ⊢ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ V
7		sseq1 3938	. 2 ⊢ (𝑧 = ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) → (𝑧 ⊆ 𝐵 ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵))
8		sseq1 3938	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵))
9		sseq1 3938	. 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
10		ssdifss 4079	. . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∖ 𝑦) ⊆ 𝐵)
11		ssdifss 4079	. . 3 ⊢ (𝑦 ⊆ 𝐵 → (𝑦 ∖ 𝑥) ⊆ 𝐵)
12		unss 4191	. . . 4 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
13	12	biimpi 219	. . 3 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
14	10, 11, 13	syl2an 608	. 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
15	1, 6, 7, 8, 9, 14	cllem0 44710	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ 𝐴

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {cab 2741  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-uni 4871

This theorem is used by: (None)