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Mirrors > Home > ILE Home > Th. List > undifss | GIF version |
Description: Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Ref | Expression |
---|---|
undifss | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3110 | . . . 4 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵 | |
2 | 1 | jctr 308 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵)) |
3 | unss 3158 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) | |
4 | 2, 3 | sylib 120 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) |
5 | ssun1 3147 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) | |
6 | sstr 3018 | . . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) ∧ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
7 | 5, 6 | mpan 415 | . 2 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵 → 𝐴 ⊆ 𝐵) |
8 | 4, 7 | impbii 124 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∖ cdif 2981 ∪ cun 2982 ⊆ wss 2984 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 |
This theorem is referenced by: difsnss 3557 exmidundif 3999 undifdcss 6560 |
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