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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 3243 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | inssdif0im 3377 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 ∖ cdif 3018 ∩ cin 3020 ⊆ wss 3021 ∅c0 3310 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-dif 3023 df-in 3027 df-ss 3034 df-nul 3311 |
This theorem is referenced by: ssdifin0 3391 difdifdirss 3394 fvsnun1 5549 fvsnun2 5550 phplem2 6676 unfiin 6743 xpfi 6747 sbthlem7 6779 sbthlemi8 6780 exmidfodomrlemim 6966 fihashssdif 10405 zfz1isolem1 10424 fsumlessfi 11068 setsfun 11776 setsfun0 11777 setsslid 11791 |
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