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Mirrors > Home > ILE Home > Th. List > eldifbd | GIF version |
Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
eldifbd.1 | ⊢ (φ → A ∈ (B ∖ 𝐶)) |
Ref | Expression |
---|---|
eldifbd | ⊢ (φ → ¬ A ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifbd.1 | . . 3 ⊢ (φ → A ∈ (B ∖ 𝐶)) | |
2 | eldif 2921 | . . 3 ⊢ (A ∈ (B ∖ 𝐶) ↔ (A ∈ B ∧ ¬ A ∈ 𝐶)) | |
3 | 1, 2 | sylib 127 | . 2 ⊢ (φ → (A ∈ B ∧ ¬ A ∈ 𝐶)) |
4 | 3 | simprd 107 | 1 ⊢ (φ → ¬ A ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∈ wcel 1390 ∖ cdif 2908 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-dif 2914 |
This theorem is referenced by: (None) |
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