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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 4868 | . . 3 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | syl6eq 2874 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) |
4 | n0 4312 | . . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) | |
5 | elxp3 5620 | . . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
6 | elssuni 4870 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ⊆ ∪ (𝐴 × 𝐵)) | |
7 | vex 3499 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
8 | vex 3499 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opnzi 5368 | . . . . . . . . 9 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
10 | ssn0 4356 | . . . . . . . . 9 ⊢ ((〈𝑥, 𝑦〉 ⊆ ∪ (𝐴 × 𝐵) ∧ 〈𝑥, 𝑦〉 ≠ ∅) → ∪ (𝐴 × 𝐵) ≠ ∅) | |
11 | 6, 9, 10 | sylancl 588 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
12 | 11 | adantl 484 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅) |
13 | 12 | exlimivv 1933 | . . . . . 6 ⊢ (∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅) |
14 | 5, 13 | sylbi 219 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
15 | 14 | exlimiv 1931 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
16 | 4, 15 | sylbi 219 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅) |
17 | 16 | necon4i 3053 | . 2 ⊢ (∪ (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅) |
18 | 3, 17 | impbii 211 | 1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 ∅c0 4293 〈cop 4575 ∪ cuni 4840 × cxp 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-opab 5131 df-xp 5563 |
This theorem is referenced by: rankxpsuc 9313 |
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