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| Mirrors > Home > MPE Home > Th. List > xpnz | Structured version Visualization version GIF version | ||
| Description: The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.) | 
| Ref | Expression | 
|---|---|
| xpnz | ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | n0 4352 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 2 | n0 4352 | . . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵) | |
| 3 | 1, 2 | anbi12i 628 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)) | 
| 4 | exdistrv 1954 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)) | |
| 5 | 3, 4 | bitr4i 278 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | 
| 6 | opex 5468 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
| 7 | eleq1 2828 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ (𝐴 × 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
| 8 | opelxp 5720 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
| 9 | 7, 8 | bitrdi 287 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | 
| 10 | 6, 9 | spcev 3605 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) | 
| 11 | n0 4352 | . . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) | |
| 12 | 10, 11 | sylibr 234 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐴 × 𝐵) ≠ ∅) | 
| 13 | 12 | exlimivv 1931 | . . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐴 × 𝐵) ≠ ∅) | 
| 14 | 5, 13 | sylbi 217 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≠ ∅) | 
| 15 | xpeq1 5698 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵)) | |
| 16 | 0xp 5783 | . . . . 5 ⊢ (∅ × 𝐵) = ∅ | |
| 17 | 15, 16 | eqtrdi 2792 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅) | 
| 18 | 17 | necon3i 2972 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅) | 
| 19 | xpeq2 5705 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
| 20 | xp0 6177 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
| 21 | 19, 20 | eqtrdi 2792 | . . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) | 
| 22 | 21 | necon3i 2972 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅) | 
| 23 | 18, 22 | jca 511 | . 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) | 
| 24 | 14, 23 | impbii 209 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 〈cop 4631 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 | 
| This theorem is referenced by: xpeq0 6179 ssxpb 6193 xp11 6194 unixpid 6303 xpexr2 7942 frxp 8152 xpfir 9301 axcc2lem 10477 axdc4lem 10496 pzriprnglem4 21496 mamufacex 22401 txindis 23643 2ndimaxp 32657 bj-xpnzex 36961 bj-1upln0 37011 bj-2upln1upl 37026 dibn0 41156 aks6d1c2lem4 42129 aks6d1c2 42132 aks6d1c6lem3 42174 | 
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