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Theorem 0nelop 4730
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 4231 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 4204 . . . . 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 2550 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  e.  { { A } ,  { A ,  B } } )
6 elpri 4040 . . 3  |-  ( (/)  e.  { { A } ,  { A ,  B } }  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
75, 6syl 16 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
82simpld 459 . . . . . 6  |-  ( (/)  e.  <. A ,  B >.  ->  A  e.  _V )
9 snnzg 4137 . . . . . 6  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
108, 9syl 16 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A }  =/=  (/) )
1110necomd 2731 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A } )
12 prnzg 4140 . . . . . 6  |-  ( A  e.  _V  ->  { A ,  B }  =/=  (/) )
138, 12syl 16 . . . . 5  |-  ( (/)  e.  <. A ,  B >.  ->  { A ,  B }  =/=  (/) )
1413necomd 2731 . . . 4  |-  ( (/)  e.  <. A ,  B >.  ->  (/)  =/=  { A ,  B } )
1511, 14jca 532 . . 3  |-  ( (/)  e.  <. A ,  B >.  ->  ( (/)  =/=  { A }  /\  (/)  =/=  { A ,  B }
) )
16 neanior 2785 . . 3  |-  ( (
(/)  =/=  { A }  /\  (/)  =/=  { A ,  B } )  <->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B } ) )
1715, 16sylib 196 . 2  |-  ( (/)  e.  <. A ,  B >.  ->  -.  ( (/)  =  { A }  \/  (/)  =  { A ,  B }
) )
187, 17pm2.65i 173 1  |-  -.  (/)  e.  <. A ,  B >.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106   (/)c0 3778   {csn 4020   {cpr 4022   <.cop 4026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027
This theorem is referenced by:  0nelelxp  5020
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