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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 3085. (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 3085 | . . 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 1481 ∖ cdif 3073 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 |
This theorem is referenced by: fimax2gtri 6803 finexdc 6804 unfidisj 6818 undifdc 6820 ssfirab 6830 fnfi 6833 iunfidisj 6842 hashunlem 10582 zfz1isolemiso 10614 fsumrelem 11272 iuncld 12323 fsumcncntop 12764 |
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