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Mirrors > Home > ILE Home > Th. List > disjdif2 | GIF version |
Description: The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.) |
Ref | Expression |
---|---|
disjdif2 | ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq2 3154 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅)) | |
2 | difin 3279 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
3 | dif0 3399 | . 2 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
4 | 1, 2, 3 | 3eqtr3g 2170 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ∖ cdif 3034 ∩ cin 3036 ∅c0 3329 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rab 2399 df-v 2659 df-dif 3039 df-in 3043 df-ss 3050 df-nul 3330 |
This theorem is referenced by: setsfun0 11835 |
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