![]() |
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 4092 | . 2 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵 | |
3 | 1, 2 | sstrdi 3957 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3908 ⊆ wss 3911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-dif 3914 df-in 3918 df-ss 3928 |
This theorem is referenced by: difss2d 4095 ssdifsn 4749 sbthlem1 9030 bcthlem2 24705 ismblfin 36165 |
Copyright terms: Public domain | W3C validator |