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Theorem 0nelop 4711
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 23 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  <. A ,  B >. )
2 oprcl 4215 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 4188 . . . . 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 2519 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  { { A } ,  { A ,  B } } )
6 elpri 4022 . . 3  |-  ( (/)  e.  { { A } ,  { A ,  B } }  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
75, 6syl 17 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
82simpld 460 . . . . . 6  |-  ( (/)  e.  <. A ,  B >.  ->  A  e.  _V )
9 snnzg 4120 . . . . . 6  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
108, 9syl 17 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A }  =/=  (/) )
1110necomd 2702 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A } )
12 prnzg 4123 . . . . . 6  |-  ( A  e.  _V  ->  { A ,  B }  =/=  (/) )
138, 12syl 17 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A ,  B }  =/=  (/) )
1413necomd 2702 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A ,  B } )
1511, 14jca 534 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =/=  { A }  /\  (/)  =/=  { A ,  B }
) )
16 neanior 2756 . . 3  |-  ( (
(/)  =/=  { A }  /\  (/)  =/=  { A ,  B } )  <->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
1715, 16sylib 199 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
187, 17pm2.65i 176 1  |-  -.  (/)  e.  <. A ,  B >.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087   (/)c0 3767   {csn 4002   {cpr 4004   <.cop 4008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009
This theorem is referenced by:  0nelelxp  4883
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