Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > difss2 | Structured version Visualization version GIF version |
Description: If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
difss2 | ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ (𝐵 ∖ 𝐶)) | |
2 | difss 3699 | . 2 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵 | |
3 | 1, 2 | syl6ss 3580 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) |
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: difss2d 3702 sbthlem1 7955 bcthlem2 22930 ismblfin 32620 |
Copyright terms: Public domain | W3C validator |