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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 257 | . 2 ⊢ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) |
4 | unss 3117 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
5 | 3, 4 | mpbi 133 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∪ cun 2915 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 |
This theorem is referenced by: undifabs 3300 inundifss 3301 dmrnssfld 4595 ltrelxr 7080 nn0ssre 8185 nn0ssz 8263 |
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