ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xp01disjl Unicode version

Theorem xp01disjl 6297
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
Assertion
Ref Expression
xp01disjl  |-  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  C ) )  =  (/)

Proof of Theorem xp01disjl
StepHypRef Expression
1 1n0 6295 . . 3  |-  1o  =/=  (/)
21necomi 2368 . 2  |-  (/)  =/=  1o
3 disjsn2 3554 . 2  |-  ( (/)  =/=  1o  ->  ( { (/)
}  i^i  { 1o } )  =  (/) )
4 xpdisj1 4931 . 2  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  ->  ( ( { (/) }  X.  A )  i^i  ( { 1o }  X.  C ) )  =  (/) )
52, 3, 4mp2b 8 1  |-  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  C ) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1314    =/= wne 2283    i^i cin 3038   (/)c0 3331   {csn 3495    X. cxp 4505   1oc1o 6272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-suc 4261  df-xp 4513  df-rel 4514  df-1o 6279
This theorem is referenced by:  djucomen  7036  djuassen  7037  xpdjuen  7038
  Copyright terms: Public domain W3C validator