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Theorem eldifeldifsn 4816
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 4813 . . 3 (𝑋𝐴 → {𝑋} ⊆ 𝐴)
21sscond 4156 . 2 (𝑋𝐴 → (𝐵𝐴) ⊆ (𝐵 ∖ {𝑋}))
32sselda 3995 1 ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  cdif 3960  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-ss 3980  df-sn 4632
This theorem is referenced by: (None)
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