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Theorem axlowdim1 25612
Description: The lower dimensional axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 25613. (Contributed by Scott Fenton, 22-Apr-2013.)
Assertion
Ref Expression
axlowdim1  |-  ( N  e.  NN  ->  E. x  e.  ( EE `  N
) E. y  e.  ( EE `  N
) x  =/=  y
)
Distinct variable group:    x, N, y

Proof of Theorem axlowdim1
StepHypRef Expression
1 1re 9023 . . . 4  |-  1  e.  RR
21fconst6 5573 . . 3  |-  ( ( 1 ... N )  X.  { 1 } ) : ( 1 ... N ) --> RR
3 elee 25547 . . 3  |-  ( N  e.  NN  ->  (
( ( 1 ... N )  X.  {
1 } )  e.  ( EE `  N
)  <->  ( ( 1 ... N )  X. 
{ 1 } ) : ( 1 ... N ) --> RR ) )
42, 3mpbiri 225 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  e.  ( EE `  N ) )
5 0re 9024 . . . 4  |-  0  e.  RR
65fconst6 5573 . . 3  |-  ( ( 1 ... N )  X.  { 0 } ) : ( 1 ... N ) --> RR
7 elee 25547 . . 3  |-  ( N  e.  NN  ->  (
( ( 1 ... N )  X.  {
0 } )  e.  ( EE `  N
)  <->  ( ( 1 ... N )  X. 
{ 0 } ) : ( 1 ... N ) --> RR ) )
86, 7mpbiri 225 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 0 } )  e.  ( EE `  N ) )
9 ax-1ne0 8992 . . . . . . 7  |-  1  =/=  0
10 df-ne 2552 . . . . . . 7  |-  ( 1  =/=  0  <->  -.  1  =  0 )
119, 10mpbi 200 . . . . . 6  |-  -.  1  =  0
12 1ex 9019 . . . . . . 7  |-  1  e.  _V
1312sneqr 3908 . . . . . 6  |-  ( { 1 }  =  {
0 }  ->  1  =  0 )
1411, 13mto 169 . . . . 5  |-  -.  {
1 }  =  {
0 }
15 elnnuz 10454 . . . . . . . . 9  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
16 eluzfz1 10996 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
1715, 16sylbi 188 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
18 ne0i 3577 . . . . . . . 8  |-  ( 1  e.  ( 1 ... N )  ->  (
1 ... N )  =/=  (/) )
1917, 18syl 16 . . . . . . 7  |-  ( N  e.  NN  ->  (
1 ... N )  =/=  (/) )
20 rnxp 5239 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 1 } )  =  { 1 } )
2119, 20syl 16 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
1 } )  =  { 1 } )
22 rnxp 5239 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 0 } )  =  { 0 } )
2319, 22syl 16 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
0 } )  =  { 0 } )
2421, 23eqeq12d 2401 . . . . 5  |-  ( N  e.  NN  ->  ( ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } )  <->  { 1 }  =  { 0 } ) )
2514, 24mtbiri 295 . . . 4  |-  ( N  e.  NN  ->  -.  ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } ) )
26 rneq 5035 . . . 4  |-  ( ( ( 1 ... N
)  X.  { 1 } )  =  ( ( 1 ... N
)  X.  { 0 } )  ->  ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } ) )
2725, 26nsyl 115 . . 3  |-  ( N  e.  NN  ->  -.  ( ( 1 ... N )  X.  {
1 } )  =  ( ( 1 ... N )  X.  {
0 } ) )
2827neneqad 2620 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  =/=  (
( 1 ... N
)  X.  { 0 } ) )
29 neeq1 2558 . . 3  |-  ( x  =  ( ( 1 ... N )  X. 
{ 1 } )  ->  ( x  =/=  y  <->  ( ( 1 ... N )  X. 
{ 1 } )  =/=  y ) )
30 neeq2 2559 . . 3  |-  ( y  =  ( ( 1 ... N )  X. 
{ 0 } )  ->  ( ( ( 1 ... N )  X.  { 1 } )  =/=  y  <->  ( (
1 ... N )  X. 
{ 1 } )  =/=  ( ( 1 ... N )  X. 
{ 0 } ) ) )
3129, 30rspc2ev 3003 . 2  |-  ( ( ( ( 1 ... N )  X.  {
1 } )  e.  ( EE `  N
)  /\  ( (
1 ... N )  X. 
{ 0 } )  e.  ( EE `  N )  /\  (
( 1 ... N
)  X.  { 1 } )  =/=  (
( 1 ... N
)  X.  { 0 } ) )  ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE
`  N ) x  =/=  y )
324, 8, 28, 31syl3anc 1184 1  |-  ( N  e.  NN  ->  E. x  e.  ( EE `  N
) E. y  e.  ( EE `  N
) x  =/=  y
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650   (/)c0 3571   {csn 3757    X. cxp 4816   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020   RRcr 8922   0cc0 8923   1c1 8924   NNcn 9932   ZZ>=cuz 10420   ...cfz 10975   EEcee 25541
This theorem is referenced by:  btwndiff  25675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-z 10215  df-uz 10421  df-fz 10976  df-ee 25544
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