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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 3141 | . . . 4 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵 | |
2 | 1 | jctr 309 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵)) |
3 | unss 3189 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) | |
4 | 2, 3 | sylib 121 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) |
5 | ssun1 3178 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) | |
6 | sstr 3047 | . . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) ∧ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
7 | 5, 6 | mpan 416 | . 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 3010 ∪ cun 3011 ⊆ wss 3013 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 |
This theorem is referenced by: difsnss 3605 exmidundif 4058 exmidundifim 4059 undifdcss 6713 |
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