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Theorem sssymdifcl 37385
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 3192 . . . 4 𝑥 ∈ V
3 difexg 4773 . . . 4 (𝑥 ∈ V → (𝑥𝑦) ∈ V)
42, 3ax-mp 5 . . 3 (𝑥𝑦) ∈ V
5 vex 3192 . . . 4 𝑦 ∈ V
6 difexg 4773 . . . 4 (𝑦 ∈ V → (𝑦𝑥) ∈ V)
75, 6ax-mp 5 . . 3 (𝑦𝑥) ∈ V
84, 7unex 6916 . 2 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ V
9 sseq1 3610 . 2 (𝑧 = ((𝑥𝑦) ∪ (𝑦𝑥)) → (𝑧𝐵 ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵))
10 sseq1 3610 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
11 sseq1 3610 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
12 ssdifss 3724 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
13 ssdifss 3724 . . 3 (𝑦𝐵 → (𝑦𝑥) ⊆ 𝐵)
14 unss 3770 . . . 4 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1514biimpi 206 . . 3 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1612, 13, 15syl2an 494 . 2 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
171, 8, 9, 10, 11, 16cllem0 37379 1 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3189  cdif 3556  cun 3557  wss 3559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-sn 4154  df-pr 4156  df-uni 4408
This theorem is referenced by: (None)
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