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Mirrors > Home > ILE Home > Th. List > disjdif | GIF version |
Description: A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.) |
Ref | Expression |
---|---|
disjdif | ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3347 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | inssdif0im 3481 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∖ cdif 3118 ∩ cin 3120 ⊆ wss 3121 ∅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-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 |
This theorem is referenced by: ssdifin0 3495 difdifdirss 3498 fvsnun1 5690 fvsnun2 5691 phplem2 6827 unfiin 6899 xpfi 6903 sbthlem7 6936 sbthlemi8 6937 exmidfodomrlemim 7165 fihashssdif 10740 zfz1isolem1 10762 fsumlessfi 11410 fprodsplit1f 11584 setsfun 12438 setsfun0 12439 setsslid 12453 |
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