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Mirrors > Home > MPE Home > Th. List > eldifsnd | Structured version Visualization version GIF version |
Description: Membership in a set with an element removed : deduction version. (Contributed by Thierry Arnoux, 4-May-2025.) |
Ref | Expression |
---|---|
eldifsnd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
eldifsnd.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Ref | Expression |
---|---|
eldifsnd | ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsnd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | eldifsnd.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
3 | eldifsn 4790 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶)) | |
4 | 1, 2, 3 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-dif 3965 df-sn 4631 |
This theorem is referenced by: prproe 4909 drngmcl 20766 r1pid2 26215 pfxchn 32983 chnind 32984 irrednzr 33236 fracfld 33289 rprmasso2 33533 rprmirredlem 33537 1arithidomlem1 33542 ufdprmidl 33548 1arithufdlem3 33553 1arithufdlem4 33554 dfufd2lem 33556 dfufd2 33557 zringfrac 33561 ply1dg1rt 33583 r1peuqusdeg1 35627 unitscyglem4 42179 domnexpgn0cl 42509 drngmullcan 42511 drngmulrcan 42512 prjspvs 42596 |
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