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Theorem 0nelop 4691
Description: A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
0nelop  |-  -.  (/)  e.  <. A ,  B >.

Proof of Theorem 0nelop
StepHypRef Expression
1 id 22 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  <. A ,  B >. )
2 oprcl 4183 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 4156 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
42, 3syl 17 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
51, 4eleqtrd 2551 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  { { A } ,  { A ,  B } } )
6 elpri 3976 . . 3  |-  ( (/)  e.  { { A } ,  { A ,  B } }  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
75, 6syl 17 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
82simpld 466 . . . . . 6  |-  ( (/)  e.  <. A ,  B >.  ->  A  e.  _V )
9 snnzg 4080 . . . . . 6  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
108, 9syl 17 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A }  =/=  (/) )
1110necomd 2698 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A } )
12 prnzg 4083 . . . . . 6  |-  ( A  e.  _V  ->  { A ,  B }  =/=  (/) )
138, 12syl 17 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A ,  B }  =/=  (/) )
1413necomd 2698 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A ,  B } )
1511, 14jca 541 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =/=  { A }  /\  (/)  =/=  { A ,  B }
) )
16 neanior 2735 . . 3  |-  ( (
(/)  =/=  { A }  /\  (/)  =/=  { A ,  B } )  <->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
1715, 16sylib 201 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
187, 17pm2.65i 178 1  |-  -.  (/)  e.  <. A ,  B >.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031   (/)c0 3722   {csn 3959   {cpr 3961   <.cop 3965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966
This theorem is referenced by:  0nelelxp  4868
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