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Mirrors > Home > ILE Home > Th. List > invdif | GIF version |
Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
invdif | ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2684 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | eldif 3075 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | mpbiran 924 | . . . 4 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵) |
4 | 3 | anbi2i 452 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
5 | elin 3254 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (V ∖ 𝐵))) | |
6 | eldif 3075 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
7 | 4, 5, 6 | 3bitr4i 211 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) |
8 | 7 | eqriv 2134 | 1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∖ cdif 3063 ∩ cin 3065 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-in 3072 |
This theorem is referenced by: indif2 3315 difundir 3324 difindir 3326 difdif2ss 3328 difun1 3331 difdifdirss 3442 nn0supp 9022 |
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