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Theorem sssymdifcl 38379
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 3343 . . . 4 𝑥 ∈ V
3 difexg 4960 . . . 4 (𝑥 ∈ V → (𝑥𝑦) ∈ V)
42, 3ax-mp 5 . . 3 (𝑥𝑦) ∈ V
5 vex 3343 . . . 4 𝑦 ∈ V
6 difexg 4960 . . . 4 (𝑦 ∈ V → (𝑦𝑥) ∈ V)
75, 6ax-mp 5 . . 3 (𝑦𝑥) ∈ V
84, 7unex 7121 . 2 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ V
9 sseq1 3767 . 2 (𝑧 = ((𝑥𝑦) ∪ (𝑦𝑥)) → (𝑧𝐵 ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵))
10 sseq1 3767 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
11 sseq1 3767 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
12 ssdifss 3884 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
13 ssdifss 3884 . . 3 (𝑦𝐵 → (𝑦𝑥) ⊆ 𝐵)
14 unss 3930 . . . 4 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1514biimpi 206 . . 3 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1612, 13, 15syl2an 495 . 2 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
171, 8, 9, 10, 11, 16cllem0 38373 1 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  Vcvv 3340  cdif 3712  cun 3713  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324  df-uni 4589
This theorem is referenced by: (None)
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