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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 3296 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶)) | |
2 | incom 3263 | . . 3 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵 ∖ 𝐶)) | |
3 | disjdif 3430 | . . 3 ⊢ (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅ | |
4 | 2, 3 | eqtri 2158 | . 2 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅ |
5 | sseq0 3399 | . 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶) ∧ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | |
6 | 1, 4, 5 | sylancl 409 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∖ cdif 3063 ∩ cin 3065 ⊆ wss 3066 ∅c0 3358 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-nul 3359 |
This theorem is referenced by: ssdifeq0 3440 |
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