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| Mirrors > Home > MPE Home > Th. List > 0nelxp | Structured version Visualization version GIF version | ||
| Description: The empty set is not a member of a Cartesian product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof shortened by JJ, 13-Aug-2021.) |
| Ref | Expression |
|---|---|
| 0nelxp | ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 2 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 3 | 1, 2 | opnzi 5479 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
| 4 | 3 | nesymi 2998 | . . . . 5 ⊢ ¬ ∅ = 〈𝑥, 𝑦〉 |
| 5 | 4 | intnanr 487 | . . . 4 ⊢ ¬ (∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
| 6 | 5 | nex 1800 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
| 7 | 6 | nex 1800 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
| 8 | elxp 5708 | . 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
| 9 | 7, 8 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∅c0 4333 〈cop 4632 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: 0nelrel0 5745 nrelv 5810 dmsn0 6229 onxpdisj 6510 mpoxopx0ov0 8241 dmtpos 8263 0nnq 10964 adderpq 10996 mulerpq 10997 lterpq 11010 0ncn 11173 structcnvcnv 17190 vtxval0 29056 iedgval0 29057 msrrcl 35548 |
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