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Theorem 0nelxp 5711
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 3479 . . . . . . 7 𝑥 ∈ V
2 vex 3479 . . . . . . 7 𝑦 ∈ V
31, 2opnzi 5475 . . . . . 6 𝑥, 𝑦⟩ ≠ ∅
43nesymi 2999 . . . . 5 ¬ ∅ = ⟨𝑥, 𝑦
54intnanr 489 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
65nex 1803 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
76nex 1803 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
8 elxp 5700 . 2 (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
97, 8mtbir 323 1 ¬ ∅ ∈ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1542  wex 1782  wcel 2107  c0 4323  cop 4635   × cxp 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683
This theorem is referenced by:  0nelrel0  5737  nrelv  5801  dmsn0  6209  onxpdisj  6491  mpoxopx0ov0  8201  dmtpos  8223  0nnq  10919  adderpq  10951  mulerpq  10952  lterpq  10965  0ncn  11128  structcnvcnv  17086  vtxval0  28299  iedgval0  28300  msrrcl  34534
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