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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 4359 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | n0 4359 | . . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵) | |
3 | 1, 2 | anbi12i 628 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)) |
4 | exdistrv 1953 | . . . 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 5475 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
7 | eleq1 2827 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ (𝐴 × 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
8 | opelxp 5725 | . . . . . . 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 3606 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) |
11 | n0 4359 | . . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) | |
12 | 10, 11 | sylibr 234 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐴 × 𝐵) ≠ ∅) |
13 | 12 | exlimivv 1930 | . . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐴 × 𝐵) ≠ ∅) |
14 | 5, 13 | sylbi 217 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≠ ∅) |
15 | xpeq1 5703 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵)) | |
16 | 0xp 5787 | . . . . 5 ⊢ (∅ × 𝐵) = ∅ | |
17 | 15, 16 | eqtrdi 2791 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅) |
18 | 17 | necon3i 2971 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅) |
19 | xpeq2 5710 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
20 | xp0 6180 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
21 | 19, 20 | eqtrdi 2791 | . . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
22 | 21 | necon3i 2971 | . . 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 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 〈cop 4637 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: xpeq0 6182 ssxpb 6196 xp11 6197 unixpid 6306 xpexr2 7942 frxp 8150 xpfir 9298 axcc2lem 10474 axdc4lem 10493 pzriprnglem4 21513 mamufacex 22416 txindis 23658 2ndimaxp 32663 bj-xpnzex 36942 bj-1upln0 36992 bj-2upln1upl 37007 dibn0 41136 aks6d1c2lem4 42109 aks6d1c2 42112 aks6d1c6lem3 42154 |
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