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Mirrors > Home > ILE Home > Th. List > difeqri | GIF version |
Description: Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
difeqri.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) |
Ref | Expression |
---|---|
difeqri | ⊢ (𝐴 ∖ 𝐵) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3030 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
2 | difeqri.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | |
3 | 1, 2 | bitri 183 | . 2 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 𝑥 ∈ 𝐶) |
4 | 3 | eqriv 2097 | 1 ⊢ (𝐴 ∖ 𝐵) = 𝐶 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 = wceq 1299 ∈ wcel 1448 ∖ cdif 3018 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-dif 3023 |
This theorem is referenced by: difdif 3148 ddifnel 3154 difab 3292 |
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