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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 3944 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
2 | df-rab 3425 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
3 | 1, 2 | eqtr4i 2755 | 1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2701 {crab 3424 ∖ cdif 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-cleq 2716 df-rab 3425 df-dif 3944 |
This theorem is referenced by: dfdif3 4107 difeq1 4108 difeq2 4109 nfdif 4118 difid 4363 ordintdif 6405 kmlem3 10144 incexc2 15786 cnambfre 37040 alephiso3 42860 sqrtcvallem1 42932 |
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