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Theorem pr2ne 8286
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 4066 . . . . 5  |-  ( B  =  A  ->  { A ,  B }  =  { A ,  A }
)
21eqcoms 2466 . . . 4  |-  ( A  =  B  ->  { A ,  B }  =  { A ,  A }
)
3 enpr1g 7488 . . . . . . . 8  |-  ( A  e.  C  ->  { A ,  A }  ~~  1o )
4 prex 4645 . . . . . . . . . . . 12  |-  { A ,  B }  e.  _V
5 eqeng 7456 . . . . . . . . . . . 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 7474 . . . . . . . . . . . . 13  |-  ( ( { A ,  B }  ~~  { A ,  A }  /\  { A ,  A }  ~~  1o )  ->  { A ,  B }  ~~  1o )
8 1sdom2 7625 . . . . . . . . . . . . . . . 16  |-  1o  ~<  2o
9 sdomnen 7451 . . . . . . . . . . . . . . . 16  |-  ( 1o 
~<  2o  ->  -.  1o  ~~  2o )
108, 9ax-mp 5 . . . . . . . . . . . . . . 15  |-  -.  1o  ~~  2o
11 ensym 7471 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  ~~  1o  ->  1o  ~~  { A ,  B }
)
12 entr 7474 . . . . . . . . . . . . . . . . 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 2665 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  ->  A  =/=  B ) )
28 pr2nelem 8285 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
29283expia 1190 . 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 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   {cpr 3990   class class class wbr 4403   1oc1o 7026   2oc2o 7027    ~~ cen 7420    ~< csdm 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-1o 7033  df-2o 7034  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426
This theorem is referenced by:  prdom2  8287  pmtrrn2  16088  mdetunilem7  18559
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