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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 2560 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 2560 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 3972 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
4 | simpl 102 | . . . . . . 7 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∅ = 〈𝑥, 𝑦〉) | |
5 | 4 | eqcomd 2045 | . . . . . 6 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 〈𝑥, 𝑦〉 = ∅) |
6 | 5 | necon3ai 2254 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ → ¬ (∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
7 | 3, 6 | ax-mp 7 | . . . 4 ⊢ ¬ (∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
8 | 7 | nex 1389 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
9 | 8 | nex 1389 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
10 | elxp 4362 | . 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
11 | 9, 10 | mtbir 596 | 1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ≠ wne 2204 ∅c0 3224 〈cop 3378 × cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 |
This theorem is referenced by: dmsn0 4788 nfunv 4933 reldmtpos 5868 dmtpos 5871 0ncn 6908 |
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