Theorem sssymdifcl 43755

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 3442	. . . 4 ⊢ 𝑥 ∈ V
3	2	difexi 5273	. . 3 ⊢ (𝑥 ∖ 𝑦) ∈ V
4		vex 3442	. . . 4 ⊢ 𝑦 ∈ V
5	4	difexi 5273	. . 3 ⊢ (𝑦 ∖ 𝑥) ∈ V
6	3, 5	unex 7687	. 2 ⊢ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ V
7		sseq1 3957	. 2 ⊢ (𝑧 = ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) → (𝑧 ⊆ 𝐵 ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵))
8		sseq1 3957	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵))
9		sseq1 3957	. 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
10		ssdifss 4090	. . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∖ 𝑦) ⊆ 𝐵)
11		ssdifss 4090	. . 3 ⊢ (𝑦 ⊆ 𝐵 → (𝑦 ∖ 𝑥) ⊆ 𝐵)
12		unss 4140	. . . 4 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
13	12	biimpi 216	. . 3 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
14	10, 11, 13	syl2an 596	. 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
15	1, 6, 7, 8, 9, 14	cllem0 43749	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ 𝐴

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  ∀wral 3049  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-sn 4579  df-pr 4581  df-uni 4862

This theorem is referenced by: (None)