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Mirrors > Home > ILE Home > Th. List > unssbd | GIF version |
Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3245. Partial converse of unssd 3247. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
unssad.1 | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Ref | Expression |
---|---|
unssbd | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unssad.1 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
2 | unss 3245 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
3 | 1, 2 | sylibr 133 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
4 | 3 | simprd 113 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∪ cun 3064 ⊆ wss 3066 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 |
This theorem is referenced by: eldifpw 4393 ertr 6437 diffifi 6781 sumsplitdc 11194 fsum2dlemstep 11196 fsumabs 11227 fsumiun 11239 |
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