Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > djussxp | Structured version Visualization version GIF version |
Description: Disjoint union is a subset of a Cartesian product. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Ref | Expression |
---|---|
djussxp | ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunss 4497 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
2 | snssi 4280 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
3 | ssv 3588 | . . 3 ⊢ 𝐵 ⊆ V | |
4 | xpss12 5148 | . . 3 ⊢ (({𝑥} ⊆ 𝐴 ∧ 𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
5 | 2, 3, 4 | sylancl 693 | . 2 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) |
6 | 1, 5 | mprgbir 2911 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 ∪ ciun 4455 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-sn 4126 df-iun 4457 df-opab 4644 df-xp 5044 |
This theorem is referenced by: djudisj 5480 iundom2g 9241 |
Copyright terms: Public domain | W3C validator |