Theorem eldifeldifsn 4772

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 4746	. . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴)
2	1	sscond 4085	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ {𝑋}))
3	2	sselda 3913	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147   ∖ cdif 4048  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-sn 4589

This theorem is used by: (None)