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Theorem 0nelxp 5722
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 3481 . . . . . . 7 𝑥 ∈ V
2 vex 3481 . . . . . . 7 𝑦 ∈ V
31, 2opnzi 5484 . . . . . 6 𝑥, 𝑦⟩ ≠ ∅
43nesymi 2995 . . . . 5 ¬ ∅ = ⟨𝑥, 𝑦
54intnanr 487 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
65nex 1796 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
76nex 1796 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
8 elxp 5711 . 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 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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