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Theorem ssdifcl 38374
Description: The class of all subsets of a class is closed under class difference. (Contributed by Richard Penner, 3-Jan-2020.)
Hypothesis
Ref Expression
ssficl.a 𝐴 = {𝑧𝑧𝐵}
Assertion
Ref Expression
ssdifcl 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴   𝑧,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐵(𝑥,𝑦)

Proof of Theorem ssdifcl
StepHypRef Expression
1 ssficl.a . 2 𝐴 = {𝑧𝑧𝐵}
2 vex 3339 . . 3 𝑥 ∈ V
32difexi 4957 . 2 (𝑥𝑦) ∈ V
4 sseq1 3763 . 2 (𝑧 = (𝑥𝑦) → (𝑧𝐵 ↔ (𝑥𝑦) ⊆ 𝐵))
5 sseq1 3763 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
6 sseq1 3763 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
7 ssdifss 3880 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
87adantr 472 . 2 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ⊆ 𝐵)
91, 3, 4, 5, 6, 8cllem0 38369 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1628  wcel 2135  {cab 2742  wral 3046  Vcvv 3336  cdif 3708  wss 3711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-v 3338  df-dif 3714  df-in 3718  df-ss 3725
This theorem is referenced by: (None)
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