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

Theorem nelpri 3642
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
Hypotheses
Ref Expression
nelpri.1  |-  A  =/= 
B
nelpri.2  |-  A  =/= 
C
Assertion
Ref Expression
nelpri  |-  -.  A  e.  { B ,  C }

Proof of Theorem nelpri
StepHypRef Expression
1 nelpri.1 . 2  |-  A  =/= 
B
2 nelpri.2 . 2  |-  A  =/= 
C
3 neanior 2451 . . 3  |-  ( ( A  =/=  B  /\  A  =/=  C )  <->  -.  ( A  =  B  \/  A  =  C )
)
4 elpri 3641 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
54con3i 633 . . 3  |-  ( -.  ( A  =  B  \/  A  =  C )  ->  -.  A  e.  { B ,  C } )
63, 5sylbi 121 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C )  ->  -.  A  e.  { B ,  C } )
71, 2, 6mp2an 426 1  |-  -.  A  e.  { B ,  C }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    \/ wo 709    = wceq 1364    e. wcel 2164    =/= wne 2364   {cpr 3619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625
This theorem is referenced by:  prneli  3643  pw1nel3  7281  sucpw1nel3  7283
  Copyright terms: Public domain W3C validator