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Mirrors > Home > ILE Home > Th. List > iunss1 | GIF version |
Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iunss1 | ⊢ (A ⊆ B → ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ B 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrexv 2999 | . . 3 ⊢ (A ⊆ B → (∃x ∈ A y ∈ 𝐶 → ∃x ∈ B y ∈ 𝐶)) | |
2 | eliun 3652 | . . 3 ⊢ (y ∈ ∪ x ∈ A 𝐶 ↔ ∃x ∈ A y ∈ 𝐶) | |
3 | eliun 3652 | . . 3 ⊢ (y ∈ ∪ x ∈ B 𝐶 ↔ ∃x ∈ B y ∈ 𝐶) | |
4 | 1, 2, 3 | 3imtr4g 194 | . 2 ⊢ (A ⊆ B → (y ∈ ∪ x ∈ A 𝐶 → y ∈ ∪ x ∈ B 𝐶)) |
5 | 4 | ssrdv 2945 | 1 ⊢ (A ⊆ B → ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ B 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1390 ∃wrex 2301 ⊆ wss 2911 ∪ ciun 3648 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-in 2918 df-ss 2925 df-iun 3650 |
This theorem is referenced by: iuneq1 3661 iunxdif2 3696 |
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