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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 4105 | . 2 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵 | |
3 | 1, 2 | sstrdi 3976 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3930 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 |
This theorem is referenced by: difss2d 4108 ssdifsn 4712 sbthlem1 8615 bcthlem2 23855 ismblfin 34814 |
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