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Theorem 0nelxp 4436
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 2618 . . . . . 6 𝑥 ∈ V
2 vex 2618 . . . . . 6 𝑦 ∈ V
31, 2opnzi 4034 . . . . 5 𝑥, 𝑦⟩ ≠ ∅
4 simpl 107 . . . . . . 7 ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ∅ = ⟨𝑥, 𝑦⟩)
54eqcomd 2090 . . . . . 6 ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ⟨𝑥, 𝑦⟩ = ∅)
65necon3ai 2300 . . . . 5 (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
73, 6ax-mp 7 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
87nex 1432 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
98nex 1432 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
10 elxp 4426 . 2 (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
119, 10mtbir 629 1 ¬ ∅ ∈ (𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102   = wceq 1287  wex 1424  wcel 1436  wne 2251  c0 3275  cop 3433   × cxp 4407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3930  ax-pow 3982  ax-pr 4008
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-opab 3874  df-xp 4415
This theorem is referenced by:  dmsn0  4860  nfunv  5008  reldmtpos  5965  dmtpos  5968  0ncn  7305
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