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Theorem eldifeldifsn 4810
Description: An element of a difference set is an element of the difference with a singleton. (Contributed by AV, 2-Jan-2022.)
Assertion
Ref Expression
eldifeldifsn ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))

Proof of Theorem eldifeldifsn
StepHypRef Expression
1 snssi 4807 . . 3 (𝑋𝐴 → {𝑋} ⊆ 𝐴)
21sscond 4137 . 2 (𝑋𝐴 → (𝐵𝐴) ⊆ (𝐵 ∖ {𝑋}))
32sselda 3978 1 ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  cdif 3941  {csn 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-dif 3947  df-in 3951  df-ss 3961  df-sn 4625
This theorem is referenced by: (None)
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