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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 2982. (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 2982 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | sylanbrc 408 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1433 ∖ cdif 2970 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 |
This theorem is referenced by: frirrg 4105 nndifsnid 6103 phpelm 6352 fidifsnid 6356 findcard2d 6375 findcard2sd 6376 diffifi 6378 unsnfidcex 6385 unsnfidcel 6386 |
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