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Theorem hash2prde 12478
Description: A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
Assertion
Ref Expression
hash2prde  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  ->  E. a E. b ( a  =/=  b  /\  V  =  { a ,  b } ) )
Distinct variable groups:    V, a,
b    W, a, b

Proof of Theorem hash2prde
StepHypRef Expression
1 hash2pr 12477 . 2  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  ->  E. a E. b  V  =  { a ,  b } )
2 equid 1740 . . . . . . 7  |-  b  =  b
3 vex 3116 . . . . . . . . 9  |-  a  e. 
_V
4 vex 3116 . . . . . . . . 9  |-  b  e. 
_V
53, 4, 4preqsn 4209 . . . . . . . 8  |-  ( { a ,  b }  =  { b }  <-> 
( a  =  b  /\  b  =  b ) )
6 eqeq2 2482 . . . . . . . . . . . 12  |-  ( { a ,  b }  =  { b }  ->  ( V  =  { a ,  b }  <->  V  =  {
b } ) )
7 fveq2 5864 . . . . . . . . . . . . . 14  |-  ( V  =  { b }  ->  ( # `  V
)  =  ( # `  { b } ) )
8 hashsng 12402 . . . . . . . . . . . . . . 15  |-  ( b  e.  _V  ->  ( # `
 { b } )  =  1 )
94, 8ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { b } )  =  1
107, 9syl6eq 2524 . . . . . . . . . . . . 13  |-  ( V  =  { b }  ->  ( # `  V
)  =  1 )
11 eqeq1 2471 . . . . . . . . . . . . . . 15  |-  ( (
# `  V )  =  2  ->  (
( # `  V )  =  1  <->  2  = 
1 ) )
12 1ne2 10744 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
13 df-ne 2664 . . . . . . . . . . . . . . . . . 18  |-  ( 1  =/=  2  <->  -.  1  =  2 )
14 pm2.21 108 . . . . . . . . . . . . . . . . . 18  |-  ( -.  1  =  2  -> 
( 1  =  2  ->  a  =/=  b
) )
1513, 14sylbi 195 . . . . . . . . . . . . . . . . 17  |-  ( 1  =/=  2  ->  (
1  =  2  -> 
a  =/=  b ) )
1612, 15ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( 1  =  2  ->  a  =/=  b )
1716eqcoms 2479 . . . . . . . . . . . . . . 15  |-  ( 2  =  1  ->  a  =/=  b )
1811, 17syl6bi 228 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  =  2  ->  (
( # `  V )  =  1  ->  a  =/=  b ) )
1918adantl 466 . . . . . . . . . . . . 13  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  -> 
( ( # `  V
)  =  1  -> 
a  =/=  b ) )
2010, 19syl5com 30 . . . . . . . . . . . 12  |-  ( V  =  { b }  ->  ( ( V  e.  W  /\  ( # `
 V )  =  2 )  ->  a  =/=  b ) )
216, 20syl6bi 228 . . . . . . . . . . 11  |-  ( { a ,  b }  =  { b }  ->  ( V  =  { a ,  b }  ->  ( ( V  e.  W  /\  ( # `  V )  =  2 )  -> 
a  =/=  b ) ) )
2221com13 80 . . . . . . . . . 10  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  -> 
( V  =  {
a ,  b }  ->  ( { a ,  b }  =  { b }  ->  a  =/=  b ) ) )
2322imp 429 . . . . . . . . 9  |-  ( ( ( V  e.  W  /\  ( # `  V
)  =  2 )  /\  V  =  {
a ,  b } )  ->  ( {
a ,  b }  =  { b }  ->  a  =/=  b
) )
2423com12 31 . . . . . . . 8  |-  ( { a ,  b }  =  { b }  ->  ( ( ( V  e.  W  /\  ( # `  V )  =  2 )  /\  V  =  { a ,  b } )  ->  a  =/=  b
) )
255, 24sylbir 213 . . . . . . 7  |-  ( ( a  =  b  /\  b  =  b )  ->  ( ( ( V  e.  W  /\  ( # `
 V )  =  2 )  /\  V  =  { a ,  b } )  ->  a  =/=  b ) )
262, 25mpan2 671 . . . . . 6  |-  ( a  =  b  ->  (
( ( V  e.  W  /\  ( # `  V )  =  2 )  /\  V  =  { a ,  b } )  ->  a  =/=  b ) )
27 ax-1 6 . . . . . 6  |-  ( a  =/=  b  ->  (
( ( V  e.  W  /\  ( # `  V )  =  2 )  /\  V  =  { a ,  b } )  ->  a  =/=  b ) )
2826, 27pm2.61ine 2780 . . . . 5  |-  ( ( ( V  e.  W  /\  ( # `  V
)  =  2 )  /\  V  =  {
a ,  b } )  ->  a  =/=  b )
29 simpr 461 . . . . 5  |-  ( ( ( V  e.  W  /\  ( # `  V
)  =  2 )  /\  V  =  {
a ,  b } )  ->  V  =  { a ,  b } )
3028, 29jca 532 . . . 4  |-  ( ( ( V  e.  W  /\  ( # `  V
)  =  2 )  /\  V  =  {
a ,  b } )  ->  ( a  =/=  b  /\  V  =  { a ,  b } ) )
3130ex 434 . . 3  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  -> 
( V  =  {
a ,  b }  ->  ( a  =/=  b  /\  V  =  { a ,  b } ) ) )
32312eximdv 1688 . 2  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  -> 
( E. a E. b  V  =  {
a ,  b }  ->  E. a E. b
( a  =/=  b  /\  V  =  {
a ,  b } ) ) )
331, 32mpd 15 1  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  ->  E. a E. b ( a  =/=  b  /\  V  =  { a ,  b } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   _Vcvv 3113   {csn 4027   {cpr 4029   ` cfv 5586   1c1 9489   2c2 10581   #chash 12369
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12370
This theorem is referenced by:  hash2prb  12479  hash2prd  12480  usgrarnedg  24060  cusgrarn  24135  frgraregord013  24795  usgedgimp  31872  usgedgimpALT  31875
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