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| Mirrors > Home > ILE Home > Th. List > ssdifin0 | GIF version | ||
| Description: A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.) |
| Ref | Expression |
|---|---|
| ssdifin0 | ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssrin 3398 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶)) | |
| 2 | incom 3365 | . . 3 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵 ∖ 𝐶)) | |
| 3 | disjdif 3533 | . . 3 ⊢ (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅ | |
| 4 | 2, 3 | eqtri 2226 | . 2 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅ |
| 5 | sseq0 3502 | . 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶) ∧ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | |
| 6 | 1, 4, 5 | sylancl 413 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 = wceq 1373 ∖ cdif 3163 ∩ cin 3165 ⊆ wss 3166 ∅c0 3460 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-v 2774 df-dif 3168 df-in 3172 df-ss 3179 df-nul 3461 |
| This theorem is referenced by: ssdifeq0 3543 |
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