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Theorem unixp0 6180
Description: A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
Assertion
Ref Expression
unixp0 ((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)

Proof of Theorem unixp0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4851 . . 3 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
2 uni0 4870 . . 3 ∅ = ∅
31, 2eqtrdi 2794 . 2 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
4 n0 4281 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5 elxp3 5649 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6 elssuni 4872 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ (𝐴 × 𝐵))
7 vex 3434 . . . . . . . . . 10 𝑥 ∈ V
8 vex 3434 . . . . . . . . . 10 𝑦 ∈ V
97, 8opnzi 5388 . . . . . . . . 9 𝑥, 𝑦⟩ ≠ ∅
10 ssn0 4335 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ⊆ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ≠ ∅) → (𝐴 × 𝐵) ≠ ∅)
116, 9, 10sylancl 586 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
1211adantl 482 . . . . . . 7 ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
1312exlimivv 1935 . . . . . 6 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
145, 13sylbi 216 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
1514exlimiv 1933 . . . 4 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
164, 15sylbi 216 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ≠ ∅)
1716necon4i 2979 . 2 ( (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
183, 17impbii 208 1 ((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  wne 2943  wss 3887  c0 4257  cop 4568   cuni 4840   × cxp 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-opab 5137  df-xp 5591
This theorem is referenced by:  rankxpsuc  9628
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