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Theorem 0nelelxp 5723
Description: A member of a Cartesian 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 5711 . 2 (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2 0nelop 5505 . . . . 5 ¬ ∅ ∈ ⟨𝑥, 𝑦
3 eleq2 2827 . . . . 5 (𝐶 = ⟨𝑥, 𝑦⟩ → (∅ ∈ 𝐶 ↔ ∅ ∈ ⟨𝑥, 𝑦⟩))
42, 3mtbiri 327 . . . 4 (𝐶 = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ 𝐶)
54adantr 480 . . 3 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ¬ ∅ ∈ 𝐶)
65exlimivv 1929 . 2 (∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ¬ ∅ ∈ 𝐶)
71, 6sylbi 217 1 (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wex 1775  wcel 2105  c0 4338  cop 4636   × cxp 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5210  df-xp 5694
This theorem is referenced by:  dmsn0el  6232  onxpdisj  6511
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