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Theorem sssymdifcl 39921
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 3496 . . . 4 𝑥 ∈ V
32difexi 5223 . . 3 (𝑥𝑦) ∈ V
4 vex 3496 . . . 4 𝑦 ∈ V
54difexi 5223 . . 3 (𝑦𝑥) ∈ V
63, 5unex 7461 . 2 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ V
7 sseq1 3990 . 2 (𝑧 = ((𝑥𝑦) ∪ (𝑦𝑥)) → (𝑧𝐵 ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵))
8 sseq1 3990 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
9 sseq1 3990 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
10 ssdifss 4110 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
11 ssdifss 4110 . . 3 (𝑦𝐵 → (𝑦𝑥) ⊆ 𝐵)
12 unss 4158 . . . 4 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1312biimpi 218 . . 3 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1410, 11, 13syl2an 597 . 2 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
151, 6, 7, 8, 9, 14cllem0 39915 1 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1531  wcel 2108  {cab 2797  wral 3136  Vcvv 3493  cdif 3931  cun 3932  wss 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-pr 4562  df-uni 4831
This theorem is referenced by: (None)
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