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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 3444 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3444 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 5331 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
4 | 3 | nesymi 3044 | . . . . 5 ⊢ ¬ ∅ = 〈𝑥, 𝑦〉 |
5 | 4 | intnanr 491 | . . . 4 ⊢ ¬ (∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
6 | 5 | nex 1802 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
7 | 6 | nex 1802 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
8 | elxp 5542 | . 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
9 | 7, 8 | mtbir 326 | 1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∅c0 4243 〈cop 4531 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 |
This theorem is referenced by: 0nelrel0 5576 nrelv 5637 dmsn0 6033 onxpdisj 6278 mpoxopx0ov0 7865 dmtpos 7887 0nnq 10335 adderpq 10367 mulerpq 10368 lterpq 10381 0ncn 10544 structcnvcnv 16489 vtxval0 26832 iedgval0 26833 msrrcl 32903 |
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