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Theorem pr2ne 7651
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 3720 . . . . 5  |-  ( B  =  A  ->  { A ,  B }  =  { A ,  A }
)
21eqcoms 2299 . . . 4  |-  ( A  =  B  ->  { A ,  B }  =  { A ,  A }
)
3 enpr1g 6943 . . . . . . . 8  |-  ( A  e.  C  ->  { A ,  A }  ~~  1o )
4 prex 4233 . . . . . . . . . . . 12  |-  { A ,  B }  e.  _V
5 eqeng 6911 . . . . . . . . . . . 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 6929 . . . . . . . . . . . . 13  |-  ( ( { A ,  B }  ~~  { A ,  A }  /\  { A ,  A }  ~~  1o )  ->  { A ,  B }  ~~  1o )
8 1sdom2 7077 . . . . . . . . . . . . . . . 16  |-  1o  ~<  2o
9 sdomnen 6906 . . . . . . . . . . . . . . . 16  |-  ( 1o 
~<  2o  ->  -.  1o  ~~  2o )
108, 9ax-mp 8 . . . . . . . . . . . . . . 15  |-  -.  1o  ~~  2o
11 ensym 6926 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  ~~  1o  ->  1o  ~~  { A ,  B }
)
12 entr 6929 . . . . . . . . . . . . . . . . 17  |-  ( ( 1o  ~~  { A ,  B }  /\  { A ,  B }  ~~  2o )  ->  1o  ~~  2o )
1312ex 423 . . . . . . . . . . . . . . . 16  |-  ( 1o 
~~  { A ,  B }  ->  ( { A ,  B }  ~~  2o  ->  1o  ~~  2o ) )
1411, 13syl 15 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  ~~  1o  ->  ( { A ,  B }  ~~  2o  ->  1o  ~~  2o ) )
1510, 14mtoi 169 . . . . . . . . . . . . . 14  |-  ( { A ,  B }  ~~  1o  ->  -.  { A ,  B }  ~~  2o )
1615a1d 22 . . . . . . . . . . . . 13  |-  ( { A ,  B }  ~~  1o  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) )
177, 16syl 15 . . . . . . . . . . . 12  |-  ( ( { A ,  B }  ~~  { A ,  A }  /\  { A ,  A }  ~~  1o )  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) )
1817ex 423 . . . . . . . . . . 11  |-  ( { A ,  B }  ~~  { A ,  A }  ->  ( { A ,  A }  ~~  1o  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
196, 18syl 15 . . . . . . . . . 10  |-  ( { A ,  B }  =  { A ,  A }  ->  ( { A ,  A }  ~~  1o  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
2019com12 27 . . . . . . . . 9  |-  ( { A ,  A }  ~~  1o  ->  ( { A ,  B }  =  { A ,  A }  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
2120a1dd 42 . . . . . . . 8  |-  ( { A ,  A }  ~~  1o  ->  ( { A ,  B }  =  { A ,  A }  ->  ( B  e.  D  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) ) )
223, 21syl 15 . . . . . . 7  |-  ( A  e.  C  ->  ( { A ,  B }  =  { A ,  A }  ->  ( B  e.  D  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) ) )
2322com23 72 . . . . . 6  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( { A ,  B }  =  { A ,  A }  ->  (
( A  e.  C  /\  B  e.  D
)  ->  -.  { A ,  B }  ~~  2o ) ) ) )
2423imp 418 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  =  { A ,  A }  ->  ( ( A  e.  C  /\  B  e.  D )  ->  -.  { A ,  B }  ~~  2o ) ) )
2524pm2.43a 45 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  =  { A ,  A }  ->  -.  { A ,  B }  ~~  2o ) )
262, 25syl5 28 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  -.  { A ,  B }  ~~  2o ) )
2726necon2ad 2507 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  ->  A  =/=  B ) )
28 pr2nelem 7650 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
29283expia 1153 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =/=  B  ->  { A ,  B }  ~~  2o ) )
3027, 29impbid 183 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   {cpr 3654   class class class wbr 4039   1oc1o 6488   2oc2o 6489    ~~ cen 6876    ~< csdm 6878
This theorem is referenced by:  prdom2  7652
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-2o 6496  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
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