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Theorem pr2ne 8395
Description: If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
Assertion
Ref Expression
pr2ne  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  <->  A  =/=  B ) )

Proof of Theorem pr2ne
StepHypRef Expression
1 preq2 4113 . . . . 5  |-  ( B  =  A  ->  { A ,  B }  =  { A ,  A }
)
21eqcoms 2479 . . . 4  |-  ( A  =  B  ->  { A ,  B }  =  { A ,  A }
)
3 enpr1g 7593 . . . . . . . 8  |-  ( A  e.  C  ->  { A ,  A }  ~~  1o )
4 prex 4695 . . . . . . . . . . . 12  |-  { A ,  B }  e.  _V
5 eqeng 7561 . . . . . . . . . . . 12  |-  ( { A ,  B }  e.  _V  ->  ( { A ,  B }  =  { A ,  A }  ->  { A ,  B }  ~~  { A ,  A } ) )
64, 5ax-mp 5 . . . . . . . . . . 11  |-  ( { A ,  B }  =  { A ,  A }  ->  { A ,  B }  ~~  { A ,  A } )
7 entr 7579 . . . . . . . . . . . . 13  |-  ( ( { A ,  B }  ~~  { A ,  A }  /\  { A ,  A }  ~~  1o )  ->  { A ,  B }  ~~  1o )
8 1sdom2 7730 . . . . . . . . . . . . . . . 16  |-  1o  ~<  2o
9 sdomnen 7556 . . . . . . . . . . . . . . . 16  |-  ( 1o 
~<  2o  ->  -.  1o  ~~  2o )
108, 9ax-mp 5 . . . . . . . . . . . . . . 15  |-  -.  1o  ~~  2o
11 ensym 7576 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  ~~  1o  ->  1o  ~~  { A ,  B }
)
12 entr 7579 . . . . . . . . . . . . . . . . 17  |-  ( ( 1o  ~~  { A ,  B }  /\  { A ,  B }  ~~  2o )  ->  1o  ~~  2o )
1312ex 434 . . . . . . . . . . . . . . . 16  |-  ( 1o 
~~  { A ,  B }  ->  ( { A ,  B }  ~~  2o  ->  1o  ~~  2o ) )
1411, 13syl 16 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  ~~  1o  ->  ( { A ,  B }  ~~  2o  ->  1o  ~~  2o ) )
1510, 14mtoi 178 . . . . . . . . . . . . . 14  |-  ( { A ,  B }  ~~  1o  ->  -.  { A ,  B }  ~~  2o )
1615a1d 25 . . . . . . . . . . . . 13  |-  ( { A ,  B }  ~~  1o  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) )
177, 16syl 16 . . . . . . . . . . . 12  |-  ( ( { A ,  B }  ~~  { A ,  A }  /\  { A ,  A }  ~~  1o )  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) )
1817ex 434 . . . . . . . . . . 11  |-  ( { A ,  B }  ~~  { A ,  A }  ->  ( { A ,  A }  ~~  1o  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
196, 18syl 16 . . . . . . . . . 10  |-  ( { A ,  B }  =  { A ,  A }  ->  ( { A ,  A }  ~~  1o  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
2019com12 31 . . . . . . . . 9  |-  ( { A ,  A }  ~~  1o  ->  ( { A ,  B }  =  { A ,  A }  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
2120a1dd 46 . . . . . . . 8  |-  ( { A ,  A }  ~~  1o  ->  ( { A ,  B }  =  { A ,  A }  ->  ( B  e.  D  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) ) )
223, 21syl 16 . . . . . . 7  |-  ( A  e.  C  ->  ( { A ,  B }  =  { A ,  A }  ->  ( B  e.  D  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) ) )
2322com23 78 . . . . . 6  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( { A ,  B }  =  { A ,  A }  ->  (
( A  e.  C  /\  B  e.  D
)  ->  -.  { A ,  B }  ~~  2o ) ) ) )
2423imp 429 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  =  { A ,  A }  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
2524pm2.43a 49 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  =  { A ,  A }  ->  -.  { A ,  B }  ~~  2o ) )
262, 25syl5 32 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  -.  { A ,  B }  ~~  2o ) )
2726necon2ad 2680 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  ->  A  =/=  B ) )
28 pr2nelem 8394 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
29283expia 1198 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =/=  B  ->  { A ,  B }  ~~  2o ) )
3027, 29impbid 191 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  <->  A  =/=  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118   {cpr 4035   class class class wbr 4453   1oc1o 7135   2oc2o 7136    ~~ cen 7525    ~< csdm 7527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-2o 7143  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531
This theorem is referenced by:  prdom2  8396  pmtrrn2  16358  mdetunilem7  18989
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