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Theorem axlowdim1 23995
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 23996. (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 8833 . . . 4  |-  1  e.  RR
21fconst6 5397 . . 3  |-  ( ( 1 ... N )  X.  { 1 } ) : ( 1 ... N ) --> RR
3 elee 23930 . . 3  |-  ( N  e.  NN  ->  (
( ( 1 ... N )  X.  {
1 } )  e.  ( EE `  N
)  <->  ( ( 1 ... N )  X. 
{ 1 } ) : ( 1 ... N ) --> RR ) )
42, 3mpbiri 226 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  e.  ( EE `  N ) )
5 0re 8834 . . . 4  |-  0  e.  RR
65fconst6 5397 . . 3  |-  ( ( 1 ... N )  X.  { 0 } ) : ( 1 ... N ) --> RR
7 elee 23930 . . 3  |-  ( N  e.  NN  ->  (
( ( 1 ... N )  X.  {
0 } )  e.  ( EE `  N
)  <->  ( ( 1 ... N )  X. 
{ 0 } ) : ( 1 ... N ) --> RR ) )
86, 7mpbiri 226 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 0 } )  e.  ( EE `  N ) )
9 ax-1ne0 8802 . . . . . . 7  |-  1  =/=  0
10 df-ne 2450 . . . . . . 7  |-  ( 1  =/=  0  <->  -.  1  =  0 )
119, 10mpbi 201 . . . . . 6  |-  -.  1  =  0
12 1ex 8829 . . . . . . 7  |-  1  e.  _V
1312sneqr 3782 . . . . . 6  |-  ( { 1 }  =  {
0 }  ->  1  =  0 )
1411, 13mto 169 . . . . 5  |-  -.  {
1 }  =  {
0 }
15 elnnuz 10260 . . . . . . . . 9  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
16 eluzfz1 10798 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
1715, 16sylbi 189 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
18 ne0i 3463 . . . . . . . 8  |-  ( 1  e.  ( 1 ... N )  ->  (
1 ... N )  =/=  (/) )
1917, 18syl 17 . . . . . . 7  |-  ( N  e.  NN  ->  (
1 ... N )  =/=  (/) )
20 rnxp 5106 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 1 } )  =  { 1 } )
2119, 20syl 17 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
1 } )  =  { 1 } )
22 rnxp 5106 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 0 } )  =  { 0 } )
2319, 22syl 17 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
0 } )  =  { 0 } )
2421, 23eqeq12d 2299 . . . . 5  |-  ( N  e.  NN  ->  ( ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } )  <->  { 1 }  =  { 0 } ) )
2514, 24mtbiri 296 . . . 4  |-  ( N  e.  NN  ->  -.  ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } ) )
26 rneq 4904 . . . 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 } ) )
28 df-ne 2450 . . 3  |-  ( ( ( 1 ... N
)  X.  { 1 } )  =/=  (
( 1 ... N
)  X.  { 0 } )  <->  -.  (
( 1 ... N
)  X.  { 1 } )  =  ( ( 1 ... N
)  X.  { 0 } ) )
2927, 28sylibr 205 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  =/=  (
( 1 ... N
)  X.  { 0 } ) )
30 neeq1 2456 . . 3  |-  ( x  =  ( ( 1 ... N )  X. 
{ 1 } )  ->  ( x  =/=  y  <->  ( ( 1 ... N )  X. 
{ 1 } )  =/=  y ) )
31 neeq2 2457 . . 3  |-  ( y  =  ( ( 1 ... N )  X. 
{ 0 } )  ->  ( ( ( 1 ... N )  X.  { 1 } )  =/=  y  <->  ( (
1 ... N )  X. 
{ 1 } )  =/=  ( ( 1 ... N )  X. 
{ 0 } ) ) )
3230, 31rspc2ev 2894 . 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 )
334, 8, 29, 32syl3anc 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 5    -> wi 6    = wceq 1624    e. wcel 1685    =/= wne 2448   E.wrex 2546   (/)c0 3457   {csn 3642    X. cxp 4687   ran crn 4690   -->wf 5218   ` cfv 5222  (class class class)co 5820   RRcr 8732   0cc0 8733   1c1 8734   NNcn 9742   ZZ>=cuz 10226   ...cfz 10777   EEcee 23924
This theorem is referenced by:  btwndiff  24058
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-z 10021  df-uz 10227  df-fz 10778  df-ee 23927
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