Theorem sssymdifcl 44025

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 3435	. . . 4 ⊢ 𝑥 ∈ V
3	2	difexi 5259	. . 3 ⊢ (𝑥 ∖ 𝑦) ∈ V
4		vex 3435	. . . 4 ⊢ 𝑦 ∈ V
5	4	difexi 5259	. . 3 ⊢ (𝑦 ∖ 𝑥) ∈ V
6	3, 5	unex 7688	. 2 ⊢ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ V
7		sseq1 3940	. 2 ⊢ (𝑧 = ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) → (𝑧 ⊆ 𝐵 ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵))
8		sseq1 3940	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵))
9		sseq1 3940	. 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
10		ssdifss 4071	. . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∖ 𝑦) ⊆ 𝐵)
11		ssdifss 4071	. . 3 ⊢ (𝑦 ⊆ 𝐵 → (𝑦 ∖ 𝑥) ⊆ 𝐵)
12		unss 4120	. . . 4 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) ↔ ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
13	12	biimpi 217	. . 3 ⊢ (((𝑥 ∖ 𝑦) ⊆ 𝐵 ∧ (𝑦 ∖ 𝑥) ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
14	10, 11, 13	syl2an 602	. 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ⊆ 𝐵)
15	1, 6, 7, 8, 9, 14	cllem0 44019	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ 𝐴

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  Vcvv 3431   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-sn 4557  df-pr 4559  df-uni 4840

This theorem is referenced by: (None)