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Mirrors > Home > ILE Home > Th. List > ssun | GIF version |
Description: A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.) |
Ref | Expression |
---|---|
ssun | ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun3 3287 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | |
2 | ssun4 3288 | . 2 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | |
3 | 1, 2 | jaoi 706 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 ∪ cun 3114 ⊆ wss 3116 |
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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 |
This theorem is referenced by: pwunss 4261 pwssunim 4262 |
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