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Theorem axlowdim1 25810
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 25811. (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 9054 . . . 4  |-  1  e.  RR
21fconst6 5600 . . 3  |-  ( ( 1 ... N )  X.  { 1 } ) : ( 1 ... N ) --> RR
3 elee 25745 . . 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 9055 . . . 4  |-  0  e.  RR
65fconst6 5600 . . 3  |-  ( ( 1 ... N )  X.  { 0 } ) : ( 1 ... N ) --> RR
7 elee 25745 . . 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 9023 . . . . . . 7  |-  1  =/=  0
10 df-ne 2577 . . . . . . 7  |-  ( 1  =/=  0  <->  -.  1  =  0 )
119, 10mpbi 200 . . . . . 6  |-  -.  1  =  0
12 1ex 9050 . . . . . . 7  |-  1  e.  _V
1312sneqr 3934 . . . . . 6  |-  ( { 1 }  =  {
0 }  ->  1  =  0 )
1411, 13mto 169 . . . . 5  |-  -.  {
1 }  =  {
0 }
15 elnnuz 10486 . . . . . . . . 9  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
16 eluzfz1 11028 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
1715, 16sylbi 188 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
18 ne0i 3602 . . . . . . . 8  |-  ( 1  e.  ( 1 ... N )  ->  (
1 ... N )  =/=  (/) )
1917, 18syl 16 . . . . . . 7  |-  ( N  e.  NN  ->  (
1 ... N )  =/=  (/) )
20 rnxp 5266 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 1 } )  =  { 1 } )
2119, 20syl 16 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
1 } )  =  { 1 } )
22 rnxp 5266 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 0 } )  =  { 0 } )
2319, 22syl 16 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
0 } )  =  { 0 } )
2421, 23eqeq12d 2426 . . . . 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 5062 . . . 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 2645 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  =/=  (
( 1 ... N
)  X.  { 0 } ) )
29 neeq1 2583 . . 3  |-  ( x  =  ( ( 1 ... N )  X. 
{ 1 } )  ->  ( x  =/=  y  <->  ( ( 1 ... N )  X. 
{ 1 } )  =/=  y ) )
30 neeq2 2584 . . 3  |-  ( y  =  ( ( 1 ... N )  X. 
{ 0 } )  ->  ( ( ( 1 ... N )  X.  { 1 } )  =/=  y  <->  ( (
1 ... N )  X. 
{ 1 } )  =/=  ( ( 1 ... N )  X. 
{ 0 } ) ) )
3129, 30rspc2ev 3028 . 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 1721    =/= wne 2575   E.wrex 2675   (/)c0 3596   {csn 3782    X. cxp 4843   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954   1c1 8955   NNcn 9964   ZZ>=cuz 10452   ...cfz 11007   EEcee 25739
This theorem is referenced by:  btwndiff  25873
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-z 10247  df-uz 10453  df-fz 11008  df-ee 25742
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