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Mirrors > Home > ILE Home > Th. List > unssi | GIF version |
Description: An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
unssi.1 | ⊢ 𝐴 ⊆ 𝐶 |
unssi.2 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
unssi | ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unssi.1 | . . 3 ⊢ 𝐴 ⊆ 𝐶 | |
2 | unssi.2 | . . 3 ⊢ 𝐵 ⊆ 𝐶 | |
3 | 1, 2 | pm3.2i 270 | . 2 ⊢ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) |
4 | unss 3245 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
5 | 3, 4 | mpbi 144 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: ∧ 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: undifabs 3434 inundifss 3435 dmrnssfld 4797 caserel 6965 ltrelxr 7818 nn0ssre 8974 nn0ssz 9065 strleun 12037 |
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