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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 3113 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
2 | df-rab 2451 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
3 | 1, 2 | eqtr4i 2188 | 1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1342 ∈ wcel 2135 {cab 2150 {crab 2446 ∖ cdif 3108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-4 1497 ax-17 1513 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-cleq 2157 df-rab 2451 df-dif 3113 |
This theorem is referenced by: dfdif3 3227 difeq1 3228 difeq2 3229 nfdif 3238 difidALT 3473 |
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