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Theorem sssymdifcl 38660
Description: The class of all subsets of a class is closed under symmetric difference. (Contributed by Richard Penner, 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 3388 . . . 4 𝑥 ∈ V
3 difexg 5003 . . . 4 (𝑥 ∈ V → (𝑥𝑦) ∈ V)
42, 3ax-mp 5 . . 3 (𝑥𝑦) ∈ V
5 vex 3388 . . . 4 𝑦 ∈ V
6 difexg 5003 . . . 4 (𝑦 ∈ V → (𝑦𝑥) ∈ V)
75, 6ax-mp 5 . . 3 (𝑦𝑥) ∈ V
84, 7unex 7190 . 2 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ V
9 sseq1 3822 . 2 (𝑧 = ((𝑥𝑦) ∪ (𝑦𝑥)) → (𝑧𝐵 ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵))
10 sseq1 3822 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
11 sseq1 3822 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
12 ssdifss 3939 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
13 ssdifss 3939 . . 3 (𝑦𝐵 → (𝑦𝑥) ⊆ 𝐵)
14 unss 3985 . . . 4 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1514biimpi 208 . . 3 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1612, 13, 15syl2an 590 . 2 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
171, 8, 9, 10, 11, 16cllem0 38654 1 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wcel 2157  {cab 2785  wral 3089  Vcvv 3385  cdif 3766  cun 3767  wss 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-sn 4369  df-pr 4371  df-uni 4629
This theorem is referenced by: (None)
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