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Theorem 0nelop 4272
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 19 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  <. A ,  B >. )
2 oprcl 3836 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 3810 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
42, 3syl 15 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
51, 4eleqtrd 2372 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  { { A } ,  { A ,  B } } )
6 elpri 3673 . . 3  |-  ( (/)  e.  { { A } ,  { A ,  B } }  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
75, 6syl 15 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
82simpld 445 . . . . . 6  |-  ( (/)  e.  <. A ,  B >.  ->  A  e.  _V )
9 snnzg 3756 . . . . . 6  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
108, 9syl 15 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A }  =/=  (/) )
1110necomd 2542 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A } )
12 prnzg 3759 . . . . . 6  |-  ( A  e.  _V  ->  { A ,  B }  =/=  (/) )
138, 12syl 15 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A ,  B }  =/=  (/) )
1413necomd 2542 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A ,  B } )
1511, 14jca 518 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =/=  { A }  /\  (/)  =/=  { A ,  B }
) )
16 neanior 2544 . . 3  |-  ( (
(/)  =/=  { A }  /\  (/)  =/=  { A ,  B } )  <->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
1715, 16sylib 188 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
187, 17pm2.65i 165 1  |-  -.  (/)  e.  <. A ,  B >.
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   {csn 3653   {cpr 3654   <.cop 3656
This theorem is referenced by:  0nelelxp  4734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662
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