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Theorem 0nelelxp 4615
Description: A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
Assertion
Ref Expression
0nelelxp (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)

Proof of Theorem 0nelelxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4603 . 2 (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2 0nelop 4208 . . . 4 ¬ ∅ ∈ ⟨𝑥, 𝑦
3 simpl 108 . . . . 5 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = ⟨𝑥, 𝑦⟩)
43eleq2d 2227 . . . 4 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → (∅ ∈ 𝐶 ↔ ∅ ∈ ⟨𝑥, 𝑦⟩))
52, 4mtbiri 665 . . 3 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ¬ ∅ ∈ 𝐶)
65exlimivv 1876 . 2 (∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ¬ ∅ ∈ 𝐶)
71, 6sylbi 120 1 (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1335  wex 1472  wcel 2128  c0 3394  cop 3563   × cxp 4584
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-xp 4592
This theorem is referenced by:  dmsn0el  5055
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