![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > unixp0 | Structured version Visualization version GIF version |
Description: A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.) |
Ref | Expression |
---|---|
unixp0 | ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4920 | . . 3 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅) | |
2 | uni0 4939 | . . 3 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | eqtrdi 2781 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) |
4 | n0 4346 | . . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) | |
5 | elxp3 5744 | . . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
6 | elssuni 4941 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ⊆ ∪ (𝐴 × 𝐵)) | |
7 | vex 3465 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
8 | vex 3465 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opnzi 5476 | . . . . . . . . 9 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
10 | ssn0 4402 | . . . . . . . . 9 ⊢ ((〈𝑥, 𝑦〉 ⊆ ∪ (𝐴 × 𝐵) ∧ 〈𝑥, 𝑦〉 ≠ ∅) → ∪ (𝐴 × 𝐵) ≠ ∅) | |
11 | 6, 9, 10 | sylancl 584 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
12 | 11 | adantl 480 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅) |
13 | 12 | exlimivv 1927 | . . . . . 6 ⊢ (∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅) |
14 | 5, 13 | sylbi 216 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
15 | 14 | exlimiv 1925 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
16 | 4, 15 | sylbi 216 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅) |
17 | 16 | necon4i 2965 | . 2 ⊢ (∪ (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅) |
18 | 3, 17 | impbii 208 | 1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2929 ⊆ wss 3944 ∅c0 4322 〈cop 4636 ∪ cuni 4909 × cxp 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-opab 5212 df-xp 5684 |
This theorem is referenced by: rankxpsuc 9907 |
Copyright terms: Public domain | W3C validator |