![]() |
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 4137. (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 4137 | . 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 3944 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-dif 3950 df-in 3954 df-ss 3964 |
This theorem is referenced by: ssdif2d 4142 eldifeldifsn 4813 fin23lem26 10322 isercoll2 15619 fctop 22727 ntrss 22779 iunconnlem 23151 clsconn 23154 regr1lem 23463 blcld 24234 rrxdstprj1 25157 voliunlem1 25299 elrspunidl 32820 carsgclctunlem2 33616 salexct 45348 meaiininclem 45500 carageniuncllem2 45536 seposep 47645 |
Copyright terms: Public domain | W3C validator |