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Theorem hashgt12el2 12448
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 12406 . . . 4  |-  ( # `  (/) )  =  0
2 fveq2 5866 . . . 4  |-  ( (/)  =  V  ->  ( # `  (/) )  =  (
# `  V )
)
31, 2syl5eqr 2522 . . 3  |-  ( (/)  =  V  ->  0  =  ( # `  V
) )
4 breq2 4451 . . . . . . 7  |-  ( (
# `  V )  =  0  ->  (
1  <  ( # `  V
)  <->  1  <  0
) )
54biimpd 207 . . . . . 6  |-  ( (
# `  V )  =  0  ->  (
1  <  ( # `  V
)  ->  1  <  0 ) )
65eqcoms 2479 . . . . 5  |-  ( 0  =  ( # `  V
)  ->  ( 1  <  ( # `  V
)  ->  1  <  0 ) )
7 0le1 10077 . . . . . 6  |-  0  <_  1
8 0re 9597 . . . . . . . 8  |-  0  e.  RR
9 1re 9596 . . . . . . . 8  |-  1  e.  RR
108, 9lenlti 9705 . . . . . . 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 1018 . . 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 2664 . . . 4  |-  ( (/)  =/=  V  <->  -.  (/)  =  V )
18 necom 2736 . . . 4  |-  ( (/)  =/=  V  <->  V  =/=  (/) )
1917, 18bitr3i 251 . . 3  |-  ( -.  (/)  =  V  <->  V  =/=  (/) )
20 ralnex 2910 . . . . . . . . . 10  |-  ( A. b  e.  V  -.  A  =/=  b  <->  -.  E. b  e.  V  A  =/=  b )
21 nne 2668 . . . . . . . . . . . 12  |-  ( -.  A  =/=  b  <->  A  =  b )
22 eqcom 2476 . . . . . . . . . . . 12  |-  ( A  =  b  <->  b  =  A )
2321, 22bitri 249 . . . . . . . . . . 11  |-  ( -.  A  =/=  b  <->  b  =  A )
2423ralbii 2895 . . . . . . . . . 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 4188 . . . . . . . . . . . . . 14  |-  ( V  =/=  (/)  ->  ( V  =  { A }  <->  A. b  e.  V  b  =  A ) )
2726bicomd 201 . . . . . . . . . . . . 13  |-  ( V  =/=  (/)  ->  ( A. b  e.  V  b  =  A  <->  V  =  { A } ) )
2827adantl 466 . . . . . . . . . . . 12  |-  ( ( V  e.  W  /\  V  =/=  (/) )  ->  ( A. b  e.  V  b  =  A  <->  V  =  { A } ) )
2928adantr 465 . . . . . . . . . . 11  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( A. b  e.  V  b  =  A  <-> 
V  =  { A } ) )
30 hashsnlei 12444 . . . . . . . . . . . . . 14  |-  ( { A }  e.  Fin  /\  ( # `  { A } )  <_  1
)
3130simpri 462 . . . . . . . . . . . . 13  |-  ( # `  { A } )  <_  1
32 fveq2 5866 . . . . . . . . . . . . . . 15  |-  ( V  =  { A }  ->  ( # `  V
)  =  ( # `  { A } ) )
3332breq1d 4457 . . . . . . . . . . . . . 14  |-  ( V  =  { A }  ->  ( ( # `  V
)  <_  1  <->  ( # `  { A } )  <_  1
) )
3433adantl 466 . . . . . . . . . . . . 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 434 . . . . . . . . . . 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 12398 . . . . . . . . . . . . 13  |-  ( V  e.  W  ->  ( # `
 V )  e. 
RR* )
3938adantr 465 . . . . . . . . . . . 12  |-  ( ( V  e.  W  /\  V  =/=  (/) )  ->  ( # `
 V )  e. 
RR* )
4039adantr 465 . . . . . . . . . . 11  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( # `  V
)  e.  RR* )
419rexri 9647 . . . . . . . . . . 11  |-  1  e.  RR*
42 xrlenlt 9653 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  RR*  /\  1  e.  RR* )  ->  (
( # `  V )  <_  1  <->  -.  1  <  ( # `  V
) ) )
4340, 41, 42sylancl 662 . . . . . . . . . 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 604 . . . . . 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 1190 . . . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   (/)c0 3785   {csn 4027   class class class wbr 4447   ` cfv 5588   Fincfn 7517   0cc0 9493   1c1 9494   RR*cxr 9628    < clt 9629    <_ cle 9630   #chash 12374
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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-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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-hash 12375
This theorem is referenced by:  3cyclfrgrarn  24786
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