MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unixp0 Structured version   Visualization version   GIF version

Theorem unixp0 6121
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 4835 . . 3 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
2 uni0 4852 . . 3 ∅ = ∅
31, 2syl6eq 2875 . 2 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
4 n0 4293 . . . 4 ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5 elxp3 5605 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6 elssuni 4854 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ (𝐴 × 𝐵))
7 vex 3483 . . . . . . . . . 10 𝑥 ∈ V
8 vex 3483 . . . . . . . . . 10 𝑦 ∈ V
97, 8opnzi 5353 . . . . . . . . 9 𝑥, 𝑦⟩ ≠ ∅
10 ssn0 4337 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ⊆ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ≠ ∅) → (𝐴 × 𝐵) ≠ ∅)
116, 9, 10sylancl 589 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
1211adantl 485 . . . . . . 7 ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
1312exlimivv 1934 . . . . . 6 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
145, 13sylbi 220 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
1514exlimiv 1932 . . . 4 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) ≠ ∅)
164, 15sylbi 220 . . 3 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ≠ ∅)
1716necon4i 3049 . 2 ( (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
183, 17impbii 212 1 ((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  wne 3014  wss 3919  c0 4276  cop 4556   cuni 4824   × cxp 5540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-opab 5115  df-xp 5548
This theorem is referenced by:  rankxpsuc  9308
  Copyright terms: Public domain W3C validator