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Theorem 0nelxp 5177
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 3234 . . . . . . 7 𝑥 ∈ V
2 vex 3234 . . . . . . 7 𝑦 ∈ V
31, 2opnzi 4972 . . . . . 6 𝑥, 𝑦⟩ ≠ ∅
43nesymi 2880 . . . . 5 ¬ ∅ = ⟨𝑥, 𝑦
54intnanr 981 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
65nex 1771 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
76nex 1771 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
8 elxp 5165 . 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 1523  wex 1744  wcel 2030  c0 3948  cop 4216   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149
This theorem is referenced by:  0nelrel  5196  nrelv  5277  dmsn0  5637  onxpdisj  5885  nfunv  5959  mpt2xopx0ov0  7387  reldmtpos  7405  dmtpos  7409  0nnq  9784  adderpq  9816  mulerpq  9817  lterpq  9830  0ncn  9992  structcnvcnv  15918  vtxval0  25976  iedgval0  25977  msrrcl  31566  relintabex  38204
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