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Mirrors > Home > MPE Home > Th. List > dfdif2 | Structured version Visualization version 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 3939 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
2 | df-rab 3147 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
3 | 1, 2 | eqtr4i 2847 | 1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 {crab 3142 ∖ cdif 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-cleq 2814 df-rab 3147 df-dif 3939 |
This theorem is referenced by: dfdif3 4091 difeq1 4092 difeq2 4093 nfdif 4102 difidALT 4331 ordintdif 6235 kmlem3 9572 incexc2 15187 cnambfre 34934 alephiso3 39911 |
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