Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  0nelxp Structured version   Visualization version   GIF version

Theorem 0nelxp 5067
 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.)
Assertion
Ref Expression
0nelxp ¬ ∅ ∈ (𝐴 × 𝐵)

Proof of Theorem 0nelxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . . . . 7 𝑥 ∈ V
2 vex 3176 . . . . . . 7 𝑦 ∈ V
31, 2opnzi 4869 . . . . . 6 𝑥, 𝑦⟩ ≠ ∅
43nesymi 2839 . . . . 5 ¬ ∅ = ⟨𝑥, 𝑦
54intnanr 952 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
65nex 1722 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
76nex 1722 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
8 elxp 5055 . 2 (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
97, 8mtbir 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
 Copyright terms: Public domain W3C validator