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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 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 4869 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
4 | 3 | nesymi 2839 | . . . . 5 ⊢ ¬ ∅ = 〈𝑥, 𝑦〉 |
5 | 4 | intnanr 952 | . . . 4 ⊢ ¬ (∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
6 | 5 | nex 1722 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
7 | 6 | nex 1722 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
8 | elxp 5055 | . 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
9 | 7, 8 | mtbir 312 | 1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∅c0 3874 〈cop 4131 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 |
This theorem is referenced by: dmsn0 5520 onxpdisj 5764 nfunv 5835 mpt2xopx0ov0 7229 reldmtpos 7247 dmtpos 7251 0nnq 9625 adderpq 9657 mulerpq 9658 lterpq 9671 0ncn 9833 structcnvcnv 15706 vtxval0 25714 iedgval0 25715 msrrcl 30694 relintabex 36906 |
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