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Mirrors > Home > ILE Home > Th. List > difss2 | 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 19 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ (𝐵 ∖ 𝐶)) | |
2 | difss 3112 | . 2 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵 | |
3 | 1, 2 | syl6ss 3024 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∖ cdif 2983 ⊆ wss 2986 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-v 2616 df-dif 2988 df-in 2992 df-ss 2999 |
This theorem is referenced by: difss2d 3115 ssdifsn 3545 sbthlem1 6587 |
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