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Mirrors > Home > MPE Home > Th. List > ssdifin0 | Structured version Visualization version 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 4250 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶)) | |
2 | disjdifr 4479 | . 2 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅ | |
3 | sseq0 4409 | . 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶) ∧ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | |
4 | 1, 2, 3 | sylancl 586 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 |
This theorem is referenced by: ssdifeq0 4493 marypha1lem 9471 numacn 10087 mreexexlem2d 17690 mreexexlem4d 17692 nrmsep2 23380 isnrm3 23383 |
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