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Theorem 0nelop 4387
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 20 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  <. A ,  B >. )
2 oprcl 3950 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 3924 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
42, 3syl 16 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
51, 4eleqtrd 2463 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  { { A } ,  { A ,  B } } )
6 elpri 3777 . . 3  |-  ( (/)  e.  { { A } ,  { A ,  B } }  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
75, 6syl 16 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
82simpld 446 . . . . . 6  |-  ( (/)  e.  <. A ,  B >.  ->  A  e.  _V )
9 snnzg 3864 . . . . . 6  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
108, 9syl 16 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A }  =/=  (/) )
1110necomd 2633 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A } )
12 prnzg 3867 . . . . . 6  |-  ( A  e.  _V  ->  { A ,  B }  =/=  (/) )
138, 12syl 16 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A ,  B }  =/=  (/) )
1413necomd 2633 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A ,  B } )
1511, 14jca 519 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =/=  { A }  /\  (/)  =/=  { A ,  B }
) )
16 neanior 2635 . . 3  |-  ( (
(/)  =/=  { A }  /\  (/)  =/=  { A ,  B } )  <->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
1715, 16sylib 189 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
187, 17pm2.65i 167 1  |-  -.  (/)  e.  <. A ,  B >.
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899   (/)c0 3571   {csn 3757   {cpr 3758   <.cop 3760
This theorem is referenced by:  0nelelxp  4847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766
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