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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 3261 | . . . 4 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵 | |
2 | 1 | jctr 315 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵)) |
3 | unss 3309 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) | |
4 | 2, 3 | sylib 122 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) |
5 | ssun1 3298 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) | |
6 | sstr 3163 | . . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) ∧ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
7 | 5, 6 | mpan 424 | . 2 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵 → 𝐴 ⊆ 𝐵) |
8 | 4, 7 | impbii 126 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∖ cdif 3126 ∪ cun 3127 ⊆ wss 3129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 |
This theorem is referenced by: difsnss 3737 exmidundif 4203 exmidundifim 4204 undifdcss 6915 |
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