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Theorem eldifeldifsn 4736
 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 4733 . . 3 (𝑋𝐴 → {𝑋} ⊆ 𝐴)
21sscond 4116 . 2 (𝑋𝐴 → (𝐵𝐴) ⊆ (𝐵 ∖ {𝑋}))
32sselda 3965 1 ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2107   ∖ cdif 3931  {csn 4559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-dif 3937  df-in 3941  df-ss 3950  df-sn 4560 This theorem is referenced by: (None)
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