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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 4851 | . . 3 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅) | |
2 | uni0 4870 | . . 3 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | eqtrdi 2794 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) |
4 | n0 4281 | . . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) | |
5 | elxp3 5649 | . . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
6 | elssuni 4872 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ⊆ ∪ (𝐴 × 𝐵)) | |
7 | vex 3434 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
8 | vex 3434 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opnzi 5388 | . . . . . . . . 9 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
10 | ssn0 4335 | . . . . . . . . 9 ⊢ ((〈𝑥, 𝑦〉 ⊆ ∪ (𝐴 × 𝐵) ∧ 〈𝑥, 𝑦〉 ≠ ∅) → ∪ (𝐴 × 𝐵) ≠ ∅) | |
11 | 6, 9, 10 | sylancl 586 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
12 | 11 | adantl 482 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅) |
13 | 12 | exlimivv 1935 | . . . . . 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 1933 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
16 | 4, 15 | sylbi 216 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅) |
17 | 16 | necon4i 2979 | . 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 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3887 ∅c0 4257 〈cop 4568 ∪ cuni 4840 × cxp 5583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-opab 5137 df-xp 5591 |
This theorem is referenced by: rankxpsuc 9628 |
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