Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sscond | Structured version Visualization version GIF version |
Description: If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 3706. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ssdifd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
sscond | ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdifd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sscon 3706 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3537 ⊆ wss 3540 |
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 |
This theorem is referenced by: ssdif2d 3711 fin23lem26 9030 isercoll2 14247 fctop 20618 ntrss 20669 iunconlem 21040 clscon 21043 regr1lem 21352 blcld 22120 rrxdstprj1 23000 voliunlem1 23125 carsgclctunlem2 29708 salexct 39228 meaiininclem 39376 carageniuncllem2 39412 |
Copyright terms: Public domain | W3C validator |