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Theorem dif1o 6414
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 6405 . . . 4  |-  1o  =  { (/) }
21difeq2i 3242 . . 3  |-  ( B 
\  1o )  =  ( B  \  { (/)
} )
32eleq2i 2237 . 2  |-  ( A  e.  ( B  \  1o )  <->  A  e.  ( B  \  { (/) } ) )
4 eldifsn 3708 . 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 2141    =/= wne 2340    \ cdif 3118   (/)c0 3414   {csn 3581   1oc1o 6385
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-ne 2341  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-nul 3415  df-sn 3587  df-suc 4354  df-1o 6392
This theorem is referenced by: (None)
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