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Theorem sssymdifcl 41851
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 3450 . . . 4 𝑥 ∈ V
32difexi 5286 . . 3 (𝑥𝑦) ∈ V
4 vex 3450 . . . 4 𝑦 ∈ V
54difexi 5286 . . 3 (𝑦𝑥) ∈ V
63, 5unex 7681 . 2 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ V
7 sseq1 3970 . 2 (𝑧 = ((𝑥𝑦) ∪ (𝑦𝑥)) → (𝑧𝐵 ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵))
8 sseq1 3970 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
9 sseq1 3970 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
10 ssdifss 4096 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
11 ssdifss 4096 . . 3 (𝑦𝐵 → (𝑦𝑥) ⊆ 𝐵)
12 unss 4145 . . . 4 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1312biimpi 215 . . 3 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1410, 11, 13syl2an 597 . 2 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
151, 6, 7, 8, 9, 14cllem0 41845 1 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {cab 2714  wral 3065  Vcvv 3446  cdif 3908  cun 3909  wss 3911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-sn 4588  df-pr 4590  df-uni 4867
This theorem is referenced by: (None)
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