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Mirrors > Home > MPE Home > Th. List > dfdif3 | Structured version Visualization version GIF version |
Description: Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) (Proof shortened by SN, 15-Aug-2025.) |
Ref | Expression |
---|---|
dfdif3 | ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdif2 3971 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} | |
2 | nelb 3231 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑦 ≠ 𝑥) | |
3 | necom 2991 | . . . 4 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦) | |
4 | 3 | ralbii 3090 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ≠ 𝑥 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) |
5 | 2, 4 | bitri 275 | . 2 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) |
6 | 1, 5 | rabbieq 3441 | 1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 {crab 3432 ∖ cdif 3959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-dif 3965 |
This theorem is referenced by: (None) |
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