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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 3138. (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 3138 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | sylanbrc 417 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2148 ∖ cdif 3126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 |
This theorem is referenced by: exmidundif 4203 exmidundifim 4204 frirrg 4346 dcdifsnid 6498 phpelm 6859 findcard2d 6884 findcard2sd 6885 diffifi 6887 unsnfidcex 6912 unsnfidcel 6913 undifdcss 6915 difinfsnlem 7091 difinfsn 7092 hashunlem 10755 seq3coll 10793 fsum3cvg 11357 isumss 11370 fisumss 11371 fproddccvg 11551 fprodssdc 11569 sqrt2irr0 12134 nnoddn2prmb 12232 logbgcd1irr 14018 |
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