Theorem eldifeldifsn 4744

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	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))

Proof of Theorem eldifeldifsn

Step	Hyp	Ref	Expression
1		snssi 4741	. . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴)
2	1	sscond 4076	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ {𝑋}))
3	2	sselda 3921	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   ∖ cdif 3884  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-sn 4562

This theorem is referenced by: (None)