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Theorem pr2ne 7889
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 3884 . . . . 5  |-  ( B  =  A  ->  { A ,  B }  =  { A ,  A }
)
21eqcoms 2439 . . . 4  |-  ( A  =  B  ->  { A ,  B }  =  { A ,  A }
)
3 enpr1g 7173 . . . . . . . 8  |-  ( A  e.  C  ->  { A ,  A }  ~~  1o )
4 prex 4406 . . . . . . . . . . . 12  |-  { A ,  B }  e.  _V
5 eqeng 7141 . . . . . . . . . . . 12  |-  ( { A ,  B }  e.  _V  ->  ( { A ,  B }  =  { A ,  A }  ->  { A ,  B }  ~~  { A ,  A } ) )
64, 5ax-mp 8 . . . . . . . . . . 11  |-  ( { A ,  B }  =  { A ,  A }  ->  { A ,  B }  ~~  { A ,  A } )
7 entr 7159 . . . . . . . . . . . . 13  |-  ( ( { A ,  B }  ~~  { A ,  A }  /\  { A ,  A }  ~~  1o )  ->  { A ,  B }  ~~  1o )
8 1sdom2 7307 . . . . . . . . . . . . . . . 16  |-  1o  ~<  2o
9 sdomnen 7136 . . . . . . . . . . . . . . . 16  |-  ( 1o 
~<  2o  ->  -.  1o  ~~  2o )
108, 9ax-mp 8 . . . . . . . . . . . . . . 15  |-  -.  1o  ~~  2o
11 ensym 7156 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  ~~  1o  ->  1o  ~~  { A ,  B }
)
12 entr 7159 . . . . . . . . . . . . . . . . 17  |-  ( ( 1o  ~~  { A ,  B }  /\  { A ,  B }  ~~  2o )  ->  1o  ~~  2o )
1312ex 424 . . . . . . . . . . . . . . . 16  |-  ( 1o 
~~  { A ,  B }  ->  ( { A ,  B }  ~~  2o  ->  1o  ~~  2o ) )
1411, 13syl 16 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  ~~  1o  ->  ( { A ,  B }  ~~  2o  ->  1o  ~~  2o ) )
1510, 14mtoi 171 . . . . . . . . . . . . . 14  |-  ( { A ,  B }  ~~  1o  ->  -.  { A ,  B }  ~~  2o )
1615a1d 23 . . . . . . . . . . . . 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 424 . . . . . . . . . . 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 29 . . . . . . . . 9  |-  ( { A ,  A }  ~~  1o  ->  ( { A ,  B }  =  { A ,  A }  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
2120a1dd 44 . . . . . . . 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 74 . . . . . 6  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( { A ,  B }  =  { A ,  A }  ->  (
( A  e.  C  /\  B  e.  D
)  ->  -.  { A ,  B }  ~~  2o ) ) ) )
2423imp 419 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  =  { A ,  A }  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
2524pm2.43a 47 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  =  { A ,  A }  ->  -.  { A ,  B }  ~~  2o ) )
262, 25syl5 30 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  -.  { A ,  B }  ~~  2o ) )
2726necon2ad 2652 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  ->  A  =/=  B ) )
28 pr2nelem 7888 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
29283expia 1155 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =/=  B  ->  { A ,  B }  ~~  2o ) )
3027, 29impbid 184 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  <->  A  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   {cpr 3815   class class class wbr 4212   1oc1o 6717   2oc2o 6718    ~~ cen 7106    ~< csdm 7108
This theorem is referenced by:  prdom2  7890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-2o 6725  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112
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