MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xp01disj Structured version   Unicode version

Theorem xp01disj 7138
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
Assertion
Ref Expression
xp01disj  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)

Proof of Theorem xp01disj
StepHypRef Expression
1 1n0 7137 . . 3  |-  1o  =/=  (/)
21necomi 2724 . 2  |-  (/)  =/=  1o
3 xpsndisj 5415 . 2  |-  ( (/)  =/=  1o  ->  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o } ) )  =  (/) )
42, 3ax-mp 5 1  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    =/= wne 2649    i^i cin 3460   (/)c0 3783   {csn 4016    X. cxp 4986   1oc1o 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-1o 7122
This theorem is referenced by:  endisj  7597  uncdadom  8542  cdaun  8543  cdaen  8544  cda1dif  8547  pm110.643  8548  cdacomen  8552  cdaassen  8553  xpcdaen  8554  mapcdaen  8555  cdadom1  8557  infcda1  8564
  Copyright terms: Public domain W3C validator