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Theorem pr2ne 7635
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 3707 . . . . 5  |-  ( B  =  A  ->  { A ,  B }  =  { A ,  A }
)
21eqcoms 2286 . . . 4  |-  ( A  =  B  ->  { A ,  B }  =  { A ,  A }
)
3 enpr1g 6927 . . . . . . . 8  |-  ( A  e.  C  ->  { A ,  A }  ~~  1o )
4 prex 4217 . . . . . . . . . . . 12  |-  { A ,  B }  e.  _V
5 eqeng 6895 . . . . . . . . . . . 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 6913 . . . . . . . . . . . . 13  |-  ( ( { A ,  B }  ~~  { A ,  A }  /\  { A ,  A }  ~~  1o )  ->  { A ,  B }  ~~  1o )
8 1sdom2 7061 . . . . . . . . . . . . . . . 16  |-  1o  ~<  2o
9 sdomnen 6890 . . . . . . . . . . . . . . . 16  |-  ( 1o 
~<  2o  ->  -.  1o  ~~  2o )
108, 9ax-mp 8 . . . . . . . . . . . . . . 15  |-  -.  1o  ~~  2o
11 ensym 6910 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  ~~  1o  ->  1o  ~~  { A ,  B }
)
12 entr 6913 . . . . . . . . . . . . . . . . 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 2494 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  ->  A  =/=  B ) )
28 pr2nelem 7634 . . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   {cpr 3641   class class class wbr 4023   1oc1o 6472   2oc2o 6473    ~~ cen 6860    ~< csdm 6862
This theorem is referenced by:  prdom2  7636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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