Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3080. (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 3080 | . . 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 1480 ∖ cdif 3068 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 |
This theorem is referenced by: fidifsnen 6764 fiunsnnn 6775 fimax2gtri 6795 unfidisj 6810 ssfirab 6822 fnfi 6825 iunfidisj 6834 hashunlem 10550 hashxp 10572 zfz1isolemiso 10582 fsumconst 11223 fsumrelem 11240 fsumcncntop 12725 |
Copyright terms: Public domain | W3C validator |