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Theorem sssymdifcl 43585
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 = {𝑧𝑧𝐵}
Assertion
Ref Expression
sssymdifcl 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴   𝑧,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐵(𝑥,𝑦)

Proof of Theorem sssymdifcl
StepHypRef Expression
1 ssficl.a . 2 𝐴 = {𝑧𝑧𝐵}
2 vex 3484 . . . 4 𝑥 ∈ V
32difexi 5330 . . 3 (𝑥𝑦) ∈ V
4 vex 3484 . . . 4 𝑦 ∈ V
54difexi 5330 . . 3 (𝑦𝑥) ∈ V
63, 5unex 7764 . 2 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ V
7 sseq1 4009 . 2 (𝑧 = ((𝑥𝑦) ∪ (𝑦𝑥)) → (𝑧𝐵 ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵))
8 sseq1 4009 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
9 sseq1 4009 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
10 ssdifss 4140 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
11 ssdifss 4140 . . 3 (𝑦𝐵 → (𝑦𝑥) ⊆ 𝐵)
12 unss 4190 . . . 4 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1312biimpi 216 . . 3 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1410, 11, 13syl2an 596 . 2 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
151, 6, 7, 8, 9, 14cllem0 43579 1 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  cdif 3948  cun 3949  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908
This theorem is referenced by: (None)
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