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Theorem hashgt12el2 12466
Description: In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
Assertion
Ref Expression
hashgt12el2  |-  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b )
Distinct variable groups:    V, b    A, b
Allowed substitution hint:    W( b)

Proof of Theorem hashgt12el2
StepHypRef Expression
1 hash0 12420 . . . 4  |-  ( # `  (/) )  =  0
2 fveq2 5848 . . . 4  |-  ( (/)  =  V  ->  ( # `  (/) )  =  (
# `  V )
)
31, 2syl5eqr 2509 . . 3  |-  ( (/)  =  V  ->  0  =  ( # `  V
) )
4 breq2 4443 . . . . . . 7  |-  ( (
# `  V )  =  0  ->  (
1  <  ( # `  V
)  <->  1  <  0
) )
54biimpd 207 . . . . . 6  |-  ( (
# `  V )  =  0  ->  (
1  <  ( # `  V
)  ->  1  <  0 ) )
65eqcoms 2466 . . . . 5  |-  ( 0  =  ( # `  V
)  ->  ( 1  <  ( # `  V
)  ->  1  <  0 ) )
7 0le1 10072 . . . . . 6  |-  0  <_  1
8 0re 9585 . . . . . . . 8  |-  0  e.  RR
9 1re 9584 . . . . . . . 8  |-  1  e.  RR
108, 9lenlti 9693 . . . . . . 7  |-  ( 0  <_  1  <->  -.  1  <  0 )
11 pm2.21 108 . . . . . . 7  |-  ( -.  1  <  0  -> 
( 1  <  0  ->  E. b  e.  V  A  =/=  b ) )
1210, 11sylbi 195 . . . . . 6  |-  ( 0  <_  1  ->  (
1  <  0  ->  E. b  e.  V  A  =/=  b ) )
137, 12ax-mp 5 . . . . 5  |-  ( 1  <  0  ->  E. b  e.  V  A  =/=  b )
146, 13syl6com 35 . . . 4  |-  ( 1  <  ( # `  V
)  ->  ( 0  =  ( # `  V
)  ->  E. b  e.  V  A  =/=  b ) )
15143ad2ant2 1016 . . 3  |-  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  (
0  =  ( # `  V )  ->  E. b  e.  V  A  =/=  b ) )
163, 15syl5com 30 . 2  |-  ( (/)  =  V  ->  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b ) )
17 df-ne 2651 . . . 4  |-  ( (/)  =/=  V  <->  -.  (/)  =  V )
18 necom 2723 . . . 4  |-  ( (/)  =/=  V  <->  V  =/=  (/) )
1917, 18bitr3i 251 . . 3  |-  ( -.  (/)  =  V  <->  V  =/=  (/) )
20 ralnex 2900 . . . . . . . . . 10  |-  ( A. b  e.  V  -.  A  =/=  b  <->  -.  E. b  e.  V  A  =/=  b )
21 nne 2655 . . . . . . . . . . . 12  |-  ( -.  A  =/=  b  <->  A  =  b )
22 eqcom 2463 . . . . . . . . . . . 12  |-  ( A  =  b  <->  b  =  A )
2321, 22bitri 249 . . . . . . . . . . 11  |-  ( -.  A  =/=  b  <->  b  =  A )
2423ralbii 2885 . . . . . . . . . 10  |-  ( A. b  e.  V  -.  A  =/=  b  <->  A. b  e.  V  b  =  A )
2520, 24bitr3i 251 . . . . . . . . 9  |-  ( -. 
E. b  e.  V  A  =/=  b  <->  A. b  e.  V  b  =  A )
26 eqsn 4177 . . . . . . . . . . . . . 14  |-  ( V  =/=  (/)  ->  ( V  =  { A }  <->  A. b  e.  V  b  =  A ) )
2726bicomd 201 . . . . . . . . . . . . 13  |-  ( V  =/=  (/)  ->  ( A. b  e.  V  b  =  A  <->  V  =  { A } ) )
2827adantl 464 . . . . . . . . . . . 12  |-  ( ( V  e.  W  /\  V  =/=  (/) )  ->  ( A. b  e.  V  b  =  A  <->  V  =  { A } ) )
2928adantr 463 . . . . . . . . . . 11  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( A. b  e.  V  b  =  A  <-> 
V  =  { A } ) )
30 hashsnlei 12462 . . . . . . . . . . . . . 14  |-  ( { A }  e.  Fin  /\  ( # `  { A } )  <_  1
)
3130simpri 460 . . . . . . . . . . . . 13  |-  ( # `  { A } )  <_  1
32 fveq2 5848 . . . . . . . . . . . . . . 15  |-  ( V  =  { A }  ->  ( # `  V
)  =  ( # `  { A } ) )
3332breq1d 4449 . . . . . . . . . . . . . 14  |-  ( V  =  { A }  ->  ( ( # `  V
)  <_  1  <->  ( # `  { A } )  <_  1
) )
3433adantl 464 . . . . . . . . . . . . 13  |-  ( ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  /\  V  =  { A } )  ->  ( ( # `  V )  <_  1  <->  (
# `  { A } )  <_  1
) )
3531, 34mpbiri 233 . . . . . . . . . . . 12  |-  ( ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  /\  V  =  { A } )  ->  ( # `  V
)  <_  1 )
3635ex 432 . . . . . . . . . . 11  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( V  =  { A }  ->  ( # `  V )  <_  1
) )
3729, 36sylbid 215 . . . . . . . . . 10  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( A. b  e.  V  b  =  A  ->  ( # `  V
)  <_  1 ) )
38 hashxrcl 12411 . . . . . . . . . . . . 13  |-  ( V  e.  W  ->  ( # `
 V )  e. 
RR* )
3938adantr 463 . . . . . . . . . . . 12  |-  ( ( V  e.  W  /\  V  =/=  (/) )  ->  ( # `
 V )  e. 
RR* )
4039adantr 463 . . . . . . . . . . 11  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( # `  V
)  e.  RR* )
419rexri 9635 . . . . . . . . . . 11  |-  1  e.  RR*
42 xrlenlt 9641 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  RR*  /\  1  e.  RR* )  ->  (
( # `  V )  <_  1  <->  -.  1  <  ( # `  V
) ) )
4340, 41, 42sylancl 660 . . . . . . . . . 10  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( ( # `  V
)  <_  1  <->  -.  1  <  ( # `  V
) ) )
4437, 43sylibd 214 . . . . . . . . 9  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( A. b  e.  V  b  =  A  ->  -.  1  <  (
# `  V )
) )
4525, 44syl5bi 217 . . . . . . . 8  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( -.  E. b  e.  V  A  =/=  b  ->  -.  1  <  (
# `  V )
) )
4645con4d 105 . . . . . . 7  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( 1  <  ( # `
 V )  ->  E. b  e.  V  A  =/=  b ) )
4746exp31 602 . . . . . 6  |-  ( V  e.  W  ->  ( V  =/=  (/)  ->  ( A  e.  V  ->  ( 1  <  ( # `  V
)  ->  E. b  e.  V  A  =/=  b ) ) ) )
4847com24 87 . . . . 5  |-  ( V  e.  W  ->  (
1  <  ( # `  V
)  ->  ( A  e.  V  ->  ( V  =/=  (/)  ->  E. b  e.  V  A  =/=  b ) ) ) )
49483imp 1188 . . . 4  |-  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  ( V  =/=  (/)  ->  E. b  e.  V  A  =/=  b ) )
5049com12 31 . . 3  |-  ( V  =/=  (/)  ->  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b ) )
5119, 50sylbi 195 . 2  |-  ( -.  (/)  =  V  ->  (
( V  e.  W  /\  1  <  ( # `  V )  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b ) )
5216, 51pm2.61i 164 1  |-  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   (/)c0 3783   {csn 4016   class class class wbr 4439   ` cfv 5570   Fincfn 7509   0cc0 9481   1c1 9482   RR*cxr 9616    < clt 9617    <_ cle 9618   #chash 12387
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-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-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-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  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-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388
This theorem is referenced by:  3cyclfrgrarn  25215  copisnmnd  32869
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