Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3120. (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 3120 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | sylanbrc 414 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2135 ∖ cdif 3108 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 |
This theorem is referenced by: exmidundif 4179 exmidundifim 4180 frirrg 4322 dcdifsnid 6463 phpelm 6823 findcard2d 6848 findcard2sd 6849 diffifi 6851 unsnfidcex 6876 unsnfidcel 6877 undifdcss 6879 difinfsnlem 7055 difinfsn 7056 hashunlem 10706 seq3coll 10741 fsum3cvg 11305 isumss 11318 fisumss 11319 fproddccvg 11499 fprodssdc 11517 sqrt2irr0 12073 nnoddn2prmb 12171 logbgcd1irr 13426 |
Copyright terms: Public domain | W3C validator |