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Theorem sssymdifcl 41160
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 3433 . . . 4 𝑥 ∈ V
32difexi 5250 . . 3 (𝑥𝑦) ∈ V
4 vex 3433 . . . 4 𝑦 ∈ V
54difexi 5250 . . 3 (𝑦𝑥) ∈ V
63, 5unex 7586 . 2 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ V
7 sseq1 3945 . 2 (𝑧 = ((𝑥𝑦) ∪ (𝑦𝑥)) → (𝑧𝐵 ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵))
8 sseq1 3945 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
9 sseq1 3945 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
10 ssdifss 4069 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
11 ssdifss 4069 . . 3 (𝑦𝐵 → (𝑦𝑥) ⊆ 𝐵)
12 unss 4117 . . . 4 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1312biimpi 215 . . 3 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1410, 11, 13syl2an 596 . 2 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
151, 6, 7, 8, 9, 14cllem0 41154 1 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wcel 2106  {cab 2715  wral 3064  Vcvv 3429  cdif 3883  cun 3884  wss 3886
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-sn 4562  df-pr 4564  df-uni 4840
This theorem is referenced by: (None)
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