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| Mirrors > Home > MPE Home > Th. List > difeqri | Structured version Visualization version 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 3961 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
| 2 | difeqri.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | |
| 3 | 1, 2 | bitri 275 | . 2 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 𝑥 ∈ 𝐶) |
| 4 | 3 | eqriv 2734 | 1 ⊢ (𝐴 ∖ 𝐵) = 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 |
| This theorem is referenced by: difdif 4135 ddif 4141 dfss4 4269 difin 4272 difab 4310 |
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