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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 4124 | . 2 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵 | |
3 | 1, 2 | sstrdi 3987 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3938 ⊆ wss 3941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3472 df-dif 3944 df-in 3948 df-ss 3958 |
This theorem is referenced by: difss2d 4127 ssdifsn 4781 sbthlem1 9063 bcthlem2 24766 ismblfin 36317 |
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