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Theorem unixp0 5896
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 4649 . . 3 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
2 uni0 4670 . . 3 ∅ = ∅
31, 2syl6eq 2867 . 2 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
4 n0 4143 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5 elxp3 5382 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6 elssuni 4672 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ (𝐴 × 𝐵))
7 vex 3405 . . . . . . . . . 10 𝑥 ∈ V
8 vex 3405 . . . . . . . . . 10 𝑦 ∈ V
97, 8opnzi 5145 . . . . . . . . 9 𝑥, 𝑦⟩ ≠ ∅
10 ssn0 4185 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ⊆ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ≠ ∅) → (𝐴 × 𝐵) ≠ ∅)
116, 9, 10sylancl 576 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
1211adantl 469 . . . . . . 7 ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
1312exlimivv 2023 . . . . . 6 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
145, 13sylbi 208 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
1514exlimiv 2021 . . . 4 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
164, 15sylbi 208 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ≠ ∅)
1716necon4i 3024 . 2 ( (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
183, 17impbii 200 1 ((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2157  wne 2989  wss 3780  c0 4127  cop 4387   cuni 4641   × cxp 5322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-opab 4918  df-xp 5330
This theorem is referenced by:  rankxpsuc  9001
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