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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 3080. (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 3080 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | sylanbrc 413 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ 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: exmidundif 4129 exmidundifim 4130 frirrg 4272 dcdifsnid 6400 phpelm 6760 findcard2d 6785 findcard2sd 6786 diffifi 6788 unsnfidcex 6808 unsnfidcel 6809 undifdcss 6811 difinfsnlem 6984 difinfsn 6985 hashunlem 10550 seq3coll 10585 fsum3cvg 11147 isumss 11160 fisumss 11161 fproddccvg 11341 sqrt2irr0 11842 |
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