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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 2763 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 2763 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 4264 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
4 | simpl 109 | . . . . . . 7 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∅ = 〈𝑥, 𝑦〉) | |
5 | 4 | eqcomd 2199 | . . . . . 6 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 〈𝑥, 𝑦〉 = ∅) |
6 | 5 | necon3ai 2413 | . . . . 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 1511 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
9 | 8 | nex 1511 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
10 | elxp 4676 | . 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
11 | 9, 10 | mtbir 672 | 1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 = wceq 1364 ∃wex 1503 ∈ wcel 2164 ≠ wne 2364 ∅c0 3446 〈cop 3621 × cxp 4657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-opab 4091 df-xp 4665 |
This theorem is referenced by: 0nelrel 4705 dmsn0 5133 nfunv 5287 reldmtpos 6306 dmtpos 6309 0ncn 7891 structcnvcnv 12634 |
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