Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3800 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶)) | |
2 | incom 3767 | . . 3 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵 ∖ 𝐶)) | |
3 | disjdif 3992 | . . 3 ⊢ (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅ | |
4 | 2, 3 | eqtri 2632 | . 2 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅ |
5 | sseq0 3927 | . 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶) ∧ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | |
6 | 1, 4, 5 | sylancl 693 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: ssdifeq0 4003 marypha1lem 8222 numacn 8755 mreexexlem2d 16128 mreexexlem4d 16130 nrmsep2 20970 isnrm3 20973 |
Copyright terms: Public domain | W3C validator |