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Theorem ssdifcl 39808
Description: The class of all subsets of a class is closed under class difference. (Contributed by RP, 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 3495 . . 3 𝑥 ∈ V
32difexi 5223 . 2 (𝑥𝑦) ∈ V
4 sseq1 3989 . 2 (𝑧 = (𝑥𝑦) → (𝑧𝐵 ↔ (𝑥𝑦) ⊆ 𝐵))
5 sseq1 3989 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
6 sseq1 3989 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
7 ssdifss 4109 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
87adantr 481 . 2 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ⊆ 𝐵)
91, 3, 4, 5, 6, 8cllem0 39803 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  {cab 2796  wral 3135  Vcvv 3492  cdif 3930  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949
This theorem is referenced by: (None)
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