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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 3243 | . . . 4 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵 | |
2 | 1 | jctr 313 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵)) |
3 | unss 3291 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) | |
4 | 2, 3 | sylib 121 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) |
5 | ssun1 3280 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) | |
6 | sstr 3145 | . . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) ∧ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
7 | 5, 6 | mpan 421 | . 2 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵 → 𝐴 ⊆ 𝐵) |
8 | 4, 7 | impbii 125 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∖ cdif 3108 ∪ cun 3109 ⊆ wss 3111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 |
This theorem is referenced by: difsnss 3713 exmidundif 4179 exmidundifim 4180 undifdcss 6879 |
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