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| Mirrors > Home > MPE Home > Th. List > eldifeldifsn | Structured version Visualization version GIF version | ||
| Description: An element of a difference set is an element of the difference with a singleton. (Contributed by AV, 2-Jan-2022.) | 
| Ref | Expression | 
|---|---|
| eldifeldifsn | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | snssi 4807 | . . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴) | |
| 2 | 1 | sscond 4145 | . 2 ⊢ (𝑋 ∈ 𝐴 → (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ {𝑋})) | 
| 3 | 2 | sselda 3982 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ∖ cdif 3947 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-ss 3967 df-sn 4626 | 
| This theorem is referenced by: (None) | 
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