MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashgt12el2 Structured version   Unicode version

Theorem hashgt12el2 11675
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 11638 . . . 4  |-  ( # `  (/) )  =  0
2 fveq2 5720 . . . 4  |-  ( (/)  =  V  ->  ( # `  (/) )  =  (
# `  V )
)
31, 2syl5eqr 2481 . . 3  |-  ( (/)  =  V  ->  0  =  ( # `  V
) )
4 breq2 4208 . . . . . . 7  |-  ( (
# `  V )  =  0  ->  (
1  <  ( # `  V
)  <->  1  <  0
) )
54biimpd 199 . . . . . 6  |-  ( (
# `  V )  =  0  ->  (
1  <  ( # `  V
)  ->  1  <  0 ) )
65eqcoms 2438 . . . . 5  |-  ( 0  =  ( # `  V
)  ->  ( 1  <  ( # `  V
)  ->  1  <  0 ) )
7 0le1 9543 . . . . . 6  |-  0  <_  1
8 0re 9083 . . . . . . . 8  |-  0  e.  RR
9 1re 9082 . . . . . . . 8  |-  1  e.  RR
108, 9lenlti 9185 . . . . . . 7  |-  ( 0  <_  1  <->  -.  1  <  0 )
11 pm2.21 102 . . . . . . 7  |-  ( -.  1  <  0  -> 
( 1  <  0  ->  E. b  e.  V  A  =/=  b ) )
1210, 11sylbi 188 . . . . . 6  |-  ( 0  <_  1  ->  (
1  <  0  ->  E. b  e.  V  A  =/=  b ) )
137, 12ax-mp 8 . . . . 5  |-  ( 1  <  0  ->  E. b  e.  V  A  =/=  b )
146, 13syl6com 33 . . . 4  |-  ( 1  <  ( # `  V
)  ->  ( 0  =  ( # `  V
)  ->  E. b  e.  V  A  =/=  b ) )
15143ad2ant2 979 . . 3  |-  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  (
0  =  ( # `  V )  ->  E. b  e.  V  A  =/=  b ) )
163, 15syl5com 28 . 2  |-  ( (/)  =  V  ->  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b ) )
17 df-ne 2600 . . . 4  |-  ( (/)  =/=  V  <->  -.  (/)  =  V )
18 necom 2679 . . . 4  |-  ( (/)  =/=  V  <->  V  =/=  (/) )
1917, 18bitr3i 243 . . 3  |-  ( -.  (/)  =  V  <->  V  =/=  (/) )
20 ralnex 2707 . . . . . . . . . 10  |-  ( A. b  e.  V  -.  A  =/=  b  <->  -.  E. b  e.  V  A  =/=  b )
21 nne 2602 . . . . . . . . . . . 12  |-  ( -.  A  =/=  b  <->  A  =  b )
22 eqcom 2437 . . . . . . . . . . . 12  |-  ( A  =  b  <->  b  =  A )
2321, 22bitri 241 . . . . . . . . . . 11  |-  ( -.  A  =/=  b  <->  b  =  A )
2423ralbii 2721 . . . . . . . . . 10  |-  ( A. b  e.  V  -.  A  =/=  b  <->  A. b  e.  V  b  =  A )
2520, 24bitr3i 243 . . . . . . . . 9  |-  ( -. 
E. b  e.  V  A  =/=  b  <->  A. b  e.  V  b  =  A )
26 eqsn 3952 . . . . . . . . . . . . . 14  |-  ( V  =/=  (/)  ->  ( V  =  { A }  <->  A. b  e.  V  b  =  A ) )
2726bicomd 193 . . . . . . . . . . . . 13  |-  ( V  =/=  (/)  ->  ( A. b  e.  V  b  =  A  <->  V  =  { A } ) )
2827adantl 453 . . . . . . . . . . . 12  |-  ( ( V  e.  W  /\  V  =/=  (/) )  ->  ( A. b  e.  V  b  =  A  <->  V  =  { A } ) )
2928adantr 452 . . . . . . . . . . 11  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( A. b  e.  V  b  =  A  <-> 
V  =  { A } ) )
30 hashsnlei 11672 . . . . . . . . . . . . . 14  |-  ( { A }  e.  Fin  /\  ( # `  { A } )  <_  1
)
3130simpri 449 . . . . . . . . . . . . 13  |-  ( # `  { A } )  <_  1
32 fveq2 5720 . . . . . . . . . . . . . . 15  |-  ( V  =  { A }  ->  ( # `  V
)  =  ( # `  { A } ) )
3332breq1d 4214 . . . . . . . . . . . . . 14  |-  ( V  =  { A }  ->  ( ( # `  V
)  <_  1  <->  ( # `  { A } )  <_  1
) )
3433adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  /\  V  =  { A } )  ->  ( ( # `  V )  <_  1  <->  (
# `  { A } )  <_  1
) )
3531, 34mpbiri 225 . . . . . . . . . . . 12  |-  ( ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  /\  V  =  { A } )  ->  ( # `  V
)  <_  1 )
3635ex 424 . . . . . . . . . . 11  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( V  =  { A }  ->  ( # `  V )  <_  1
) )
3729, 36sylbid 207 . . . . . . . . . 10  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( A. b  e.  V  b  =  A  ->  ( # `  V
)  <_  1 ) )
38 hashxrcl 11632 . . . . . . . . . . . . 13  |-  ( V  e.  W  ->  ( # `
 V )  e. 
RR* )
3938adantr 452 . . . . . . . . . . . 12  |-  ( ( V  e.  W  /\  V  =/=  (/) )  ->  ( # `
 V )  e. 
RR* )
4039adantr 452 . . . . . . . . . . 11  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( # `  V
)  e.  RR* )
419rexri 9129 . . . . . . . . . . 11  |-  1  e.  RR*
42 xrlenlt 9135 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  RR*  /\  1  e.  RR* )  ->  (
( # `  V )  <_  1  <->  -.  1  <  ( # `  V
) ) )
4340, 41, 42sylancl 644 . . . . . . . . . 10  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( ( # `  V
)  <_  1  <->  -.  1  <  ( # `  V
) ) )
4437, 43sylibd 206 . . . . . . . . 9  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( A. b  e.  V  b  =  A  ->  -.  1  <  (
# `  V )
) )
4525, 44syl5bi 209 . . . . . . . 8  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( -.  E. b  e.  V  A  =/=  b  ->  -.  1  <  (
# `  V )
) )
4645con4d 99 . . . . . . 7  |-  ( ( ( V  e.  W  /\  V  =/=  (/) )  /\  A  e.  V )  ->  ( 1  <  ( # `
 V )  ->  E. b  e.  V  A  =/=  b ) )
4746exp31 588 . . . . . 6  |-  ( V  e.  W  ->  ( V  =/=  (/)  ->  ( A  e.  V  ->  ( 1  <  ( # `  V
)  ->  E. b  e.  V  A  =/=  b ) ) ) )
4847com24 83 . . . . 5  |-  ( V  e.  W  ->  (
1  <  ( # `  V
)  ->  ( A  e.  V  ->  ( V  =/=  (/)  ->  E. b  e.  V  A  =/=  b ) ) ) )
49483imp 1147 . . . 4  |-  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  ( V  =/=  (/)  ->  E. b  e.  V  A  =/=  b ) )
5049com12 29 . . 3  |-  ( V  =/=  (/)  ->  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b ) )
5119, 50sylbi 188 . 2  |-  ( -.  (/)  =  V  ->  (
( V  e.  W  /\  1  <  ( # `  V )  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b ) )
5216, 51pm2.61i 158 1  |-  ( ( V  e.  W  /\  1  <  ( # `  V
)  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   (/)c0 3620   {csn 3806   class class class wbr 4204   ` cfv 5446   Fincfn 7101   0cc0 8982   1c1 8983   RR*cxr 9111    < clt 9112    <_ cle 9113   #chash 11610
This theorem is referenced by:  3cyclfrgrarn  28340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611
  Copyright terms: Public domain W3C validator