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Theorem 0nelxp 4574
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 2692 . . . . . 6 𝑥 ∈ V
2 vex 2692 . . . . . 6 𝑦 ∈ V
31, 2opnzi 4164 . . . . 5 𝑥, 𝑦⟩ ≠ ∅
4 simpl 108 . . . . . . 7 ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ∅ = ⟨𝑥, 𝑦⟩)
54eqcomd 2146 . . . . . 6 ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ⟨𝑥, 𝑦⟩ = ∅)
65necon3ai 2358 . . . . 5 (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
73, 6ax-mp 5 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
87nex 1477 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
98nex 1477 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
10 elxp 4563 . 2 (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
119, 10mtbir 661 1 ¬ ∅ ∈ (𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1332  wex 1469  wcel 1481  wne 2309  c0 3367  cop 3534   × cxp 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-opab 3997  df-xp 4552
This theorem is referenced by:  0nelrel  4592  dmsn0  5013  nfunv  5163  reldmtpos  6157  dmtpos  6160  0ncn  7662  structcnvcnv  12012
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