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Theorem unixp0 5586
 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 4380 . . 3 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
2 uni0 4401 . . 3 ∅ = ∅
31, 2syl6eq 2660 . 2 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
4 n0 3890 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5 elxp3 5092 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6 elssuni 4403 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ (𝐴 × 𝐵))
7 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
8 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
97, 8opnzi 4869 . . . . . . . . 9 𝑥, 𝑦⟩ ≠ ∅
10 ssn0 3928 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ⊆ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ≠ ∅) → (𝐴 × 𝐵) ≠ ∅)
116, 9, 10sylancl 693 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
1211adantl 481 . . . . . . 7 ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
1312exlimivv 1847 . . . . . 6 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
145, 13sylbi 206 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
1514exlimiv 1845 . . . 4 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
164, 15sylbi 206 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ≠ ∅)
1716necon4i 2817 . 2 ( (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
183, 17impbii 198 1 ((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131  ∪ cuni 4372   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-opab 4644  df-xp 5044 This theorem is referenced by:  rankxpsuc  8628
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