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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 3239 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅)) | |
2 | difin 3364 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
3 | dif0 3485 | . 2 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
4 | 1, 2, 3 | 3eqtr3g 2226 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∖ cdif 3118 ∩ cin 3120 ∅c0 3414 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rab 2457 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 |
This theorem is referenced by: setsfun0 12452 |
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