ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  int0el Unicode version

Theorem int0el 3859
Description: The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
int0el  |-  ( (/)  e.  A  ->  |^| A  =  (/) )

Proof of Theorem int0el
StepHypRef Expression
1 intss1 3844 . 2  |-  ( (/)  e.  A  ->  |^| A  C_  (/) )
2 0ss 3452 . . 3  |-  (/)  C_  |^| A
32a1i 9 . 2  |-  ( (/)  e.  A  ->  (/)  C_  |^| A
)
41, 3eqssd 3164 1  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141    C_ wss 3121   (/)c0 3414   |^|cint 3829
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-int 3830
This theorem is referenced by:  intv  4154  inton  4376
  Copyright terms: Public domain W3C validator