MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpdisj1 Structured version   Visualization version   GIF version

Theorem xpdisj1 5797
Description: Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
xpdisj1 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)

Proof of Theorem xpdisj1
StepHypRef Expression
1 xpeq1 5357 . 2 ((𝐴𝐵) = ∅ → ((𝐴𝐵) × (𝐶𝐷)) = (∅ × (𝐶𝐷)))
2 inxp 5488 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴𝐵) × (𝐶𝐷))
3 0xp 5435 . . 3 (∅ × (𝐶𝐷)) = ∅
43eqcomi 2835 . 2 ∅ = (∅ × (𝐶𝐷))
51, 2, 43eqtr4g 2887 1 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  cin 3798  c0 4145   × cxp 5341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-opab 4937  df-xp 5349  df-rel 5350
This theorem is referenced by:  djudisj  5803  djuin  9058  xpdisjres  29959  esum2dlem  30700  nosupbnd2lem1  32401  noetalem2  32404  noetalem3  32405  bj-2upln1upl  33535  disjxp1  40056
  Copyright terms: Public domain W3C validator