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Theorem ssdifcl 36895
 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 3176 . . 3 𝑥 ∈ V
3 difexg 4735 . . 3 (𝑥 ∈ V → (𝑥𝑦) ∈ V)
42, 3ax-mp 5 . 2 (𝑥𝑦) ∈ V
5 sseq1 3589 . 2 (𝑧 = (𝑥𝑦) → (𝑧𝐵 ↔ (𝑥𝑦) ⊆ 𝐵))
6 sseq1 3589 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
7 sseq1 3589 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
8 ssdifss 3703 . . 3 (𝑥𝐵 → (𝑥𝑦) ⊆ 𝐵)
98adantr 480 . 2 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ⊆ 𝐵)
101, 4, 5, 6, 7, 9cllem0 36890 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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