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Mirrors > Home > ILE Home > Th. List > eldifad | GIF version |
Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3111. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
eldifad.1 | ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
Ref | Expression |
---|---|
eldifad | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifad.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) | |
2 | eldif 3111 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
3 | 1, 2 | sylib 121 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) |
4 | 3 | simpld 111 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 2128 ∖ cdif 3099 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 |
This theorem is referenced by: fimax2gtri 6846 finexdc 6847 unfidisj 6866 undifdc 6868 ssfirab 6878 fnfi 6881 iunfidisj 6890 dcfi 6925 hashunlem 10678 zfz1isolemiso 10710 fsumrelem 11368 fprodcl2lem 11502 fprodap0 11518 fprodrec 11526 fprodap0f 11533 fprodle 11537 iuncld 12515 fsumcncntop 12956 bj-charfun 13382 |
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