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

Theorem xpdisj1 5994
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 5545 . 2 ((𝐴𝐵) = ∅ → ((𝐴𝐵) × (𝐶𝐷)) = (∅ × (𝐶𝐷)))
2 inxp 5679 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴𝐵) × (𝐶𝐷))
3 0xp 5625 . . 3 (∅ × (𝐶𝐷)) = ∅
43eqcomi 2829 . 2 ∅ = (∅ × (𝐶𝐷))
51, 2, 43eqtr4g 2880 1 ((𝐴𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3912  c0 4269   × cxp 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-opab 5105  df-xp 5537  df-rel 5538
This theorem is referenced by:  djudisj  6000  xp01disjl  8099  djuin  9325  xpdisjres  30335  esum2dlem  31359  nosupbnd2lem1  33223  noetalem2  33226  noetalem3  33227  disjxp1  41486
  Copyright terms: Public domain W3C validator