![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3947 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
2 | df-rab 3432 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} | |
3 | 1, 2 | eqtr4i 2762 | 1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2708 {crab 3431 ∖ cdif 3941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2723 df-rab 3432 df-dif 3947 |
This theorem is referenced by: dfdif3 4110 difeq1 4111 difeq2 4112 nfdif 4121 difid 4366 ordintdif 6403 kmlem3 10129 incexc2 15766 cnambfre 36338 alephiso3 42079 sqrtcvallem1 42151 |
Copyright terms: Public domain | W3C validator |