Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sssymdifcl Structured version   Visualization version   GIF version

Theorem sssymdifcl 42899
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 3472 . . . 4 𝑥 ∈ V
32difexi 5321 . . 3 (𝑥𝑦) ∈ V
4 vex 3472 . . . 4 𝑦 ∈ V
54difexi 5321 . . 3 (𝑦𝑥) ∈ V
63, 5unex 7730 . 2 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ V
7 sseq1 4002 . 2 (𝑧 = ((𝑥𝑦) ∪ (𝑦𝑥)) → (𝑧𝐵 ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵))
8 sseq1 4002 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
9 sseq1 4002 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
10 ssdifss 4130 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
11 ssdifss 4130 . . 3 (𝑦𝐵 → (𝑦𝑥) ⊆ 𝐵)
12 unss 4179 . . . 4 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) ↔ ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1312biimpi 215 . . 3 (((𝑥𝑦) ⊆ 𝐵 ∧ (𝑦𝑥) ⊆ 𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
1410, 11, 13syl2an 595 . 2 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∪ (𝑦𝑥)) ⊆ 𝐵)
151, 6, 7, 8, 9, 14cllem0 42893 1 𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  wcel 2098  {cab 2703  wral 3055  Vcvv 3468  cdif 3940  cun 3941  wss 3943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-uni 4903
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator