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Theorem 0nelelxp 5179
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 5165 . 2 (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2 0nelop 4989 . . . . 5 ¬ ∅ ∈ ⟨𝑥, 𝑦
3 eleq2 2719 . . . . 5 (𝐶 = ⟨𝑥, 𝑦⟩ → (∅ ∈ 𝐶 ↔ ∅ ∈ ⟨𝑥, 𝑦⟩))
42, 3mtbiri 316 . . . 4 (𝐶 = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ 𝐶)
54adantr 480 . . 3 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ¬ ∅ ∈ 𝐶)
65exlimivv 1900 . 2 (∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ¬ ∅ ∈ 𝐶)
71, 6sylbi 207 1 (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  c0 3948  cop 4216   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149
This theorem is referenced by:  dmsn0el  5639  onxpdisj  5885
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