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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 3370 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | inssdif0im 3505 | . 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 1364 ∖ cdif 3141 ∩ cin 3143 ⊆ wss 3144 ∅c0 3437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 |
This theorem is referenced by: ssdifin0 3519 difdifdirss 3522 fvsnun1 5733 fvsnun2 5734 phplem2 6880 unfiin 6953 xpfi 6957 sbthlem7 6991 sbthlemi8 6992 exmidfodomrlemim 7229 fihashssdif 10829 zfz1isolem1 10851 fsumlessfi 11499 fprodsplit1f 11673 setsfun 12546 setsfun0 12547 setsslid 12562 |
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