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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 4596 | . . 3 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅) | |
2 | uni0 4617 | . . 3 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | syl6eq 2810 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) |
4 | n0 4074 | . . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) | |
5 | elxp3 5326 | . . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
6 | elssuni 4619 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ⊆ ∪ (𝐴 × 𝐵)) | |
7 | vex 3343 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
8 | vex 3343 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opnzi 5091 | . . . . . . . . 9 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
10 | ssn0 4119 | . . . . . . . . 9 ⊢ ((〈𝑥, 𝑦〉 ⊆ ∪ (𝐴 × 𝐵) ∧ 〈𝑥, 𝑦〉 ≠ ∅) → ∪ (𝐴 × 𝐵) ≠ ∅) | |
11 | 6, 9, 10 | sylancl 697 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
12 | 11 | adantl 473 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅) |
13 | 12 | exlimivv 2009 | . . . . . 6 ⊢ (∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝑧 ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅) |
14 | 5, 13 | sylbi 207 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
15 | 14 | exlimiv 2007 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅) |
16 | 4, 15 | sylbi 207 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅) |
17 | 16 | necon4i 2967 | . 2 ⊢ (∪ (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅) |
18 | 3, 17 | impbii 199 | 1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1632 ∃wex 1853 ∈ wcel 2139 ≠ wne 2932 ⊆ wss 3715 ∅c0 4058 〈cop 4327 ∪ cuni 4588 × cxp 5264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-opab 4865 df-xp 5272 |
This theorem is referenced by: rankxpsuc 8918 |
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