MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  int0el Structured version   Unicode version

Theorem int0el 4284
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 4267 . 2  |-  ( (/)  e.  A  ->  |^| A  C_  (/) )
2 0ss 3791 . . 3  |-  (/)  C_  |^| A
32a1i 11 . 2  |-  ( (/)  e.  A  ->  (/)  C_  |^| A
)
41, 3eqssd 3481 1  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868    C_ wss 3436   (/)c0 3761   |^|cint 4252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-dif 3439  df-in 3443  df-ss 3450  df-nul 3762  df-int 4253
This theorem is referenced by:  intv  4597  inton  5496  onint0  6634  oev2  7230
  Copyright terms: Public domain W3C validator