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Theorem hash2prde 12503
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 12502 . 2  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  ->  E. a E. b  V  =  { a ,  b } )
2 equid 1796 . . . . . . 7  |-  b  =  b
3 vex 3109 . . . . . . . . 9  |-  a  e. 
_V
4 vex 3109 . . . . . . . . 9  |-  b  e. 
_V
53, 4, 4preqsn 4199 . . . . . . . 8  |-  ( { a ,  b }  =  { b }  <-> 
( a  =  b  /\  b  =  b ) )
6 eqeq2 2469 . . . . . . . . . . . 12  |-  ( { a ,  b }  =  { b }  ->  ( V  =  { a ,  b }  <->  V  =  {
b } ) )
7 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( V  =  { b }  ->  ( # `  V
)  =  ( # `  { b } ) )
8 hashsng 12424 . . . . . . . . . . . . . . 15  |-  ( b  e.  _V  ->  ( # `
 { b } )  =  1 )
94, 8ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { b } )  =  1
107, 9syl6eq 2511 . . . . . . . . . . . . 13  |-  ( V  =  { b }  ->  ( # `  V
)  =  1 )
11 eqeq1 2458 . . . . . . . . . . . . . . 15  |-  ( (
# `  V )  =  2  ->  (
( # `  V )  =  1  <->  2  = 
1 ) )
12 1ne2 10744 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
13 df-ne 2651 . . . . . . . . . . . . . . . . . 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 2466 . . . . . . . . . . . . . . 15  |-  ( 2  =  1  ->  a  =/=  b )
1811, 17syl6bi 228 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  =  2  ->  (
( # `  V )  =  1  ->  a  =/=  b ) )
1918adantl 464 . . . . . . . . . . . . 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 427 . . . . . . . . 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 669 . . . . . 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 2767 . . . . 5  |-  ( ( ( V  e.  W  /\  ( # `  V
)  =  2 )  /\  V  =  {
a ,  b } )  ->  a  =/=  b )
29 simpr 459 . . . . 5  |-  ( ( ( V  e.  W  /\  ( # `  V
)  =  2 )  /\  V  =  {
a ,  b } )  ->  V  =  { a ,  b } )
3028, 29jca 530 . . . 4  |-  ( ( ( V  e.  W  /\  ( # `  V
)  =  2 )  /\  V  =  {
a ,  b } )  ->  ( a  =/=  b  /\  V  =  { a ,  b } ) )
3130ex 432 . . 3  |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  -> 
( V  =  {
a ,  b }  ->  ( a  =/=  b  /\  V  =  { a ,  b } ) ) )
32312eximdv 1717 . 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 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   _Vcvv 3106   {csn 4016   {cpr 4018   ` cfv 5570   1c1 9482   2c2 10581   #chash 12390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12391
This theorem is referenced by:  hash2prb  12504  hash2prd  12505  usgrarnedg  24589  cusgrarn  24664  frgraregord013  25323  usgedgimp  32794  usgedgimpALT  32797
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