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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 4786 | . 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 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-sn 4627 |
| This theorem is referenced by: prproe 4905 drngmcl 20750 r1pid2 26201 pfxchn 32999 chnind 33001 irrednzr 33254 fracfld 33310 rprmasso2 33554 rprmirredlem 33558 1arithidomlem1 33563 ufdprmidl 33569 1arithufdlem3 33574 1arithufdlem4 33575 dfufd2lem 33577 dfufd2 33578 zringfrac 33582 ply1dg1rt 33604 r1peuqusdeg1 35648 unitscyglem4 42199 resuppsinopn 42393 readvcot 42394 domnexpgn0cl 42533 drngmullcan 42535 drngmulrcan 42536 prjspvs 42620 |
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