ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  0in Unicode version

Theorem 0in 3398
Description: The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
0in  |-  ( (/)  i^i 
A )  =  (/)

Proof of Theorem 0in
StepHypRef Expression
1 incom 3268 . 2  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
2 in0 3397 . 2  |-  ( A  i^i  (/) )  =  (/)
31, 2eqtri 2160 1  |-  ( (/)  i^i 
A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1331    i^i cin 3070   (/)c0 3363
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-nul 3364
This theorem is referenced by:  setsfun  12003  setsfun0  12004  restsn  12358
  Copyright terms: Public domain W3C validator