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

Theorem 0nelxp 5719
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 3484 . . . . . . 7 𝑥 ∈ V
2 vex 3484 . . . . . . 7 𝑦 ∈ V
31, 2opnzi 5479 . . . . . 6 𝑥, 𝑦⟩ ≠ ∅
43nesymi 2998 . . . . 5 ¬ ∅ = ⟨𝑥, 𝑦
54intnanr 487 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
65nex 1800 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
76nex 1800 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
8 elxp 5708 . 2 (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
97, 8mtbir 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
  Copyright terms: Public domain W3C validator