ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpeq0r Structured version   GIF version

Theorem xpeq0r 4689
Description: A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)
Assertion
Ref Expression
xpeq0r ((A = ∅ B = ∅) → (A × B) = ∅)

Proof of Theorem xpeq0r
StepHypRef Expression
1 xpeq1 4302 . . 3 (A = ∅ → (A × B) = (∅ × B))
2 0xp 4363 . . 3 (∅ × B) = ∅
31, 2syl6eq 2085 . 2 (A = ∅ → (A × B) = ∅)
4 xpeq2 4303 . . 3 (B = ∅ → (A × B) = (A × ∅))
5 xp0 4686 . . 3 (A × ∅) = ∅
64, 5syl6eq 2085 . 2 (B = ∅ → (A × B) = ∅)
73, 6jaoi 635 1 ((A = ∅ B = ∅) → (A × B) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242  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-bnd 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-eu 1900  df-mo 1901  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-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator