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Theorem 0nelxp 4687
Description: The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
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 2763 . . . . . 6 𝑥 ∈ V
2 vex 2763 . . . . . 6 𝑦 ∈ V
31, 2opnzi 4264 . . . . 5 𝑥, 𝑦⟩ ≠ ∅
4 simpl 109 . . . . . . 7 ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ∅ = ⟨𝑥, 𝑦⟩)
54eqcomd 2199 . . . . . 6 ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ⟨𝑥, 𝑦⟩ = ∅)
65necon3ai 2413 . . . . 5 (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
73, 6ax-mp 5 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
87nex 1511 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
98nex 1511 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
10 elxp 4676 . 2 (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
119, 10mtbir 672 1 ¬ ∅ ∈ (𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104   = wceq 1364  wex 1503  wcel 2164  wne 2364  c0 3446  cop 3621   × cxp 4657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-xp 4665
This theorem is referenced by:  0nelrel  4705  dmsn0  5133  nfunv  5287  reldmtpos  6306  dmtpos  6309  0ncn  7891  structcnvcnv  12634
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