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Mirrors > Home > ILE Home > Th. List > dfdif2 | GIF version |
Description: Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.) |
Ref | Expression |
---|---|
dfdif2 | ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dif 2920 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
2 | df-rab 2315 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
3 | 1, 2 | eqtr4i 2063 | 1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {cab 2026 {crab 2310 ∖ cdif 2914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-4 1400 ax-17 1419 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-rab 2315 df-dif 2920 |
This theorem is referenced by: difeq1 3055 difeq2 3056 nfdif 3065 difidALT 3293 |
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