ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dif1o Unicode version

Theorem dif1o 6303
Description: Two ways to say that  A is a nonzero number of the set  B. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
dif1o  |-  ( A  e.  ( B  \  1o )  <->  ( A  e.  B  /\  A  =/=  (/) ) )

Proof of Theorem dif1o
StepHypRef Expression
1 df1o2 6294 . . . 4  |-  1o  =  { (/) }
21difeq2i 3161 . . 3  |-  ( B 
\  1o )  =  ( B  \  { (/)
} )
32eleq2i 2184 . 2  |-  ( A  e.  ( B  \  1o )  <->  A  e.  ( B  \  { (/) } ) )
4 eldifsn 3620 . 2  |-  ( A  e.  ( B  \  { (/) } )  <->  ( A  e.  B  /\  A  =/=  (/) ) )
53, 4bitri 183 1  |-  ( A  e.  ( B  \  1o )  <->  ( A  e.  B  /\  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1465    =/= wne 2285    \ cdif 3038   (/)c0 3333   {csn 3497   1oc1o 6274
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-nul 3334  df-sn 3503  df-suc 4263  df-1o 6281
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator