Theorem eldifeldifsn 4811

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 4808	. . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴)
2	1	sscond 4135	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ {𝑋}))
3	2	sselda 3973	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098   ∖ cdif 3938  {csn 4625

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-dif 3944  df-ss 3958  df-sn 4626

This theorem is referenced by: (None)