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Mirrors > Home > ILE Home > Th. List > eldifd | GIF version |
Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 2927. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
eldifd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
eldifd.2 | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) |
Ref | Expression |
---|---|
eldifd | ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | eldifd.2 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) | |
3 | eldif 2927 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | sylanbrc 394 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1393 ∖ cdif 2914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 |
This theorem is referenced by: frirrg 4087 nndifsnid 6080 phpelm 6328 fidifsnid 6332 findcard2d 6348 findcard2sd 6349 diffifi 6351 |
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