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Mirrors > Home > ILE Home > Th. List > ssiun | GIF version |
Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ssiun | ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3020 | . . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵)) | |
2 | 1 | reximi 2471 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵)) |
3 | r19.37av 2521 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
5 | eliun 3740 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
6 | 4, 5 | syl6ibr 161 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
7 | 6 | ssrdv 3032 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 ∃wrex 2361 ⊆ wss 3000 ∪ ciun 3736 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-in 3006 df-ss 3013 df-iun 3738 |
This theorem is referenced by: iunss2 3781 iunpwss 3826 iunpw 4315 |
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