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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 4701 | . . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴) | |
2 | 1 | sscond 4069 | . 2 ⊢ (𝑋 ∈ 𝐴 → (𝐵 ∖ 𝐴) ⊆ (𝐵 ∖ {𝑋})) |
3 | 2 | sselda 3915 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∖ cdif 3878 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-sn 4526 |
This theorem is referenced by: (None) |
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