Theorem eldifeldifsn 4792

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 4789	. . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴)
2	1	sscond 4126	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ {𝑋}))
3	2	sselda 3963	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ∖ cdif 3928  {csn 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-dif 3934  df-ss 3948  df-sn 4607

This theorem is referenced by: (None)