Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpdisj1 Structured version   GIF version

Theorem xpdisj1 4690
 Description: Cross 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 inxp 4413 . 2 ((A × 𝐶) ∩ (B × 𝐷)) = ((AB) × (𝐶𝐷))
2 xpeq1 4302 . . 3 ((AB) = ∅ → ((AB) × (𝐶𝐷)) = (∅ × (𝐶𝐷)))
3 0xp 4363 . . 3 (∅ × (𝐶𝐷)) = ∅
42, 3syl6eq 2085 . 2 ((AB) = ∅ → ((AB) × (𝐶𝐷)) = ∅)
51, 4syl5eq 2081 1 ((AB) = ∅ → ((A × 𝐶) ∩ (B × 𝐷)) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∩ cin 2910  ∅c0 3218   × cxp 4286 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294  df-rel 4295 This theorem is referenced by:  djudisj  4693
 Copyright terms: Public domain W3C validator