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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 3481 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3481 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 5484 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
4 | 3 | nesymi 2995 | . . . . 5 ⊢ ¬ ∅ = 〈𝑥, 𝑦〉 |
5 | 4 | intnanr 487 | . . . 4 ⊢ ¬ (∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
6 | 5 | nex 1796 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
7 | 6 | nex 1796 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
8 | elxp 5711 | . 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 1536 ∃wex 1775 ∈ wcel 2105 ∅c0 4338 〈cop 4636 × cxp 5686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5210 df-xp 5694 |
This theorem is referenced by: 0nelrel0 5748 nrelv 5812 dmsn0 6230 onxpdisj 6511 mpoxopx0ov0 8239 dmtpos 8261 0nnq 10961 adderpq 10993 mulerpq 10994 lterpq 11007 0ncn 11170 structcnvcnv 17186 vtxval0 29070 iedgval0 29071 msrrcl 35527 |
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