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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 3954 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
| 2 | df-rab 3437 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
| 3 | 1, 2 | eqtr4i 2768 | 1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 {crab 3436 ∖ cdif 3948 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-rab 3437 df-dif 3954 |
| This theorem is referenced by: dfdif3 4117 dfdif3OLD 4118 difeq1 4119 difeq2 4120 nfdifOLD 4130 difid 4376 ordintdif 6434 kmlem3 10193 incexc2 15874 cnambfre 37675 alephiso3 43572 sqrtcvallem1 43644 |
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