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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 2663 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 2663 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 4127 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
4 | simpl 108 | . . . . . . 7 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∅ = 〈𝑥, 𝑦〉) | |
5 | 4 | eqcomd 2123 | . . . . . 6 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 〈𝑥, 𝑦〉 = ∅) |
6 | 5 | necon3ai 2334 | . . . . 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 1461 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
9 | 8 | nex 1461 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
10 | elxp 4526 | . 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
11 | 9, 10 | mtbir 645 | 1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1316 ∃wex 1453 ∈ wcel 1465 ≠ wne 2285 ∅c0 3333 〈cop 3500 × cxp 4507 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 |
This theorem is referenced by: 0nelrel 4555 dmsn0 4976 nfunv 5126 reldmtpos 6118 dmtpos 6121 0ncn 7607 structcnvcnv 11902 |
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