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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 4744 | . . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴) | |
2 | 1 | sscond 4121 | . 2 ⊢ (𝑋 ∈ 𝐴 → (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ {𝑋})) |
3 | 2 | sselda 3970 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ∖ cdif 3936 {csn 4570 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 df-sn 4571 |
This theorem is referenced by: (None) |
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