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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 3050. (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 3050 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
3 | 1, 2 | sylib 121 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) |
4 | 3 | simprd 113 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 1465 ∖ cdif 3038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 |
This theorem is referenced by: fidifsnen 6732 fiunsnnn 6743 fimax2gtri 6763 unfidisj 6778 ssfirab 6790 fnfi 6793 iunfidisj 6802 hashunlem 10518 hashxp 10540 zfz1isolemiso 10550 fsumconst 11191 fsumrelem 11208 fsumcncntop 12652 |
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