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Mirrors > Home > ILE Home > Th. List > 0nelxp | GIF version |
Description: The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
0nelxp | ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2724 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 2724 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 4207 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
4 | simpl 108 | . . . . . . 7 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∅ = 〈𝑥, 𝑦〉) | |
5 | 4 | eqcomd 2170 | . . . . . 6 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 〈𝑥, 𝑦〉 = ∅) |
6 | 5 | necon3ai 2383 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ → ¬ (∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
7 | 3, 6 | ax-mp 5 | . . . 4 ⊢ ¬ (∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
8 | 7 | nex 1487 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
9 | 8 | nex 1487 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
10 | elxp 4615 | . 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
11 | 9, 10 | mtbir 661 | 1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1342 ∃wex 1479 ∈ wcel 2135 ≠ wne 2334 ∅c0 3404 〈cop 3573 × cxp 4596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-xp 4604 |
This theorem is referenced by: 0nelrel 4644 dmsn0 5065 nfunv 5215 reldmtpos 6212 dmtpos 6215 0ncn 7763 structcnvcnv 12347 |
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