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Mirrors > Home > ILE Home > Th. List > eldifbd | GIF version |
Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3022. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
eldifbd.1 | ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
Ref | Expression |
---|---|
eldifbd | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifbd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) | |
2 | eldif 3022 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
3 | 1, 2 | sylib 121 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) |
4 | 3 | simprd 113 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 1445 ∖ cdif 3010 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-dif 3015 |
This theorem is referenced by: fidifsnen 6666 fiunsnnn 6677 fimax2gtri 6697 unfidisj 6712 ssfirab 6723 fnfi 6726 iunfidisj 6735 hashunlem 10327 hashxp 10349 zfz1isolemiso 10359 fsumconst 10997 fsumrelem 11014 |
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