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Mirrors > Home > ILE Home > Th. List > djussxp | GIF version |
Description: Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Ref | Expression |
---|---|
djussxp | ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunss 3854 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
2 | snssi 3664 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
3 | ssv 3119 | . . 3 ⊢ 𝐵 ⊆ V | |
4 | xpss12 4646 | . . 3 ⊢ (({𝑥} ⊆ 𝐴 ∧ 𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
5 | 2, 3, 4 | sylancl 409 | . 2 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) |
6 | 1, 5 | mprgbir 2490 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 {csn 3527 ∪ ciun 3813 × cxp 4537 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-in 3077 df-ss 3084 df-sn 3533 df-iun 3815 df-opab 3990 df-xp 4545 |
This theorem is referenced by: djudisj 4966 |
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