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Theorem dif1o 6499
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 6491 . . . 4  |-  1o  =  { (/) }
21difeq2i 3291 . . 3  |-  ( B 
\  1o )  =  ( B  \  { (/)
} )
32eleq2i 2347 . 2  |-  ( A  e.  ( B  \  1o )  <->  A  e.  ( B  \  { (/) } ) )
4 eldifsn 3749 . 2  |-  ( A  e.  ( B  \  { (/) } )  <->  ( A  e.  B  /\  A  =/=  (/) ) )
53, 4bitri 240 1  |-  ( A  e.  ( B  \  1o )  <->  ( A  e.  B  /\  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684    =/= wne 2446    \ cdif 3149   (/)c0 3455   {csn 3640   1oc1o 6472
This theorem is referenced by:  ondif1  6500  brwitnlem  6506  oelim2  6593  oeeulem  6599  oeeui  6600  omabs  6645  cantnfle  7372  cantnfp1lem2  7381  cantnfp1lem3  7382  cantnfp1  7383  oemapvali  7386  cantnflem1a  7387  cantnflem1c  7389  cantnflem1  7391  cantnflem3  7393  cantnflem4  7394  cantnf  7395  cnfcomlem  7402  cnfcom3lem  7406  cnfcom3  7407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-suc 4398  df-1o 6479
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