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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 3901 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
2 | snssi 3711 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
3 | ssv 3159 | . . 3 ⊢ 𝐵 ⊆ V | |
4 | xpss12 4705 | . . 3 ⊢ (({𝑥} ⊆ 𝐴 ∧ 𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
5 | 2, 3, 4 | sylancl 410 | . 2 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) |
6 | 1, 5 | mprgbir 2522 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 {csn 3570 ∪ ciun 3860 × cxp 4596 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-in 3117 df-ss 3124 df-sn 3576 df-iun 3862 df-opab 4038 df-xp 4604 |
This theorem is referenced by: djudisj 5025 |
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