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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 3840 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
2 | difeqri.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | |
3 | 1, 2 | bitri 267 | . 2 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 𝑥 ∈ 𝐶) |
4 | 3 | eqriv 2776 | 1 ⊢ (𝐴 ∖ 𝐵) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∖ cdif 3827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-v 3418 df-dif 3833 |
This theorem is referenced by: difdif 3998 ddif 4004 dfss4 4123 difin 4126 difab 4160 |
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