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Theorem axlowdim1 25068
Description: The lower dimension 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 25069. (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 9660 . . . 4  |-  1  e.  RR
21fconst6 5786 . . 3  |-  ( ( 1 ... N )  X.  { 1 } ) : ( 1 ... N ) --> RR
3 elee 25003 . . 3  |-  ( N  e.  NN  ->  (
( ( 1 ... N )  X.  {
1 } )  e.  ( EE `  N
)  <->  ( ( 1 ... N )  X. 
{ 1 } ) : ( 1 ... N ) --> RR ) )
42, 3mpbiri 241 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  e.  ( EE `  N ) )
5 0re 9661 . . . 4  |-  0  e.  RR
65fconst6 5786 . . 3  |-  ( ( 1 ... N )  X.  { 0 } ) : ( 1 ... N ) --> RR
7 elee 25003 . . 3  |-  ( N  e.  NN  ->  (
( ( 1 ... N )  X.  {
0 } )  e.  ( EE `  N
)  <->  ( ( 1 ... N )  X. 
{ 0 } ) : ( 1 ... N ) --> RR ) )
86, 7mpbiri 241 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 0 } )  e.  ( EE `  N ) )
9 ax-1ne0 9626 . . . . . . 7  |-  1  =/=  0
109neii 2645 . . . . . 6  |-  -.  1  =  0
11 1ex 9656 . . . . . . 7  |-  1  e.  _V
1211sneqr 4131 . . . . . 6  |-  ( { 1 }  =  {
0 }  ->  1  =  0 )
1310, 12mto 181 . . . . 5  |-  -.  {
1 }  =  {
0 }
14 elnnuz 11219 . . . . . . . . 9  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
15 eluzfz1 11832 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
1614, 15sylbi 200 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
17 ne0i 3728 . . . . . . . 8  |-  ( 1  e.  ( 1 ... N )  ->  (
1 ... N )  =/=  (/) )
1816, 17syl 17 . . . . . . 7  |-  ( N  e.  NN  ->  (
1 ... N )  =/=  (/) )
19 rnxp 5273 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 1 } )  =  { 1 } )
2018, 19syl 17 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
1 } )  =  { 1 } )
21 rnxp 5273 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 0 } )  =  { 0 } )
2218, 21syl 17 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
0 } )  =  { 0 } )
2320, 22eqeq12d 2486 . . . . 5  |-  ( N  e.  NN  ->  ( ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } )  <->  { 1 }  =  { 0 } ) )
2413, 23mtbiri 310 . . . 4  |-  ( N  e.  NN  ->  -.  ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } ) )
25 rneq 5066 . . . 4  |-  ( ( ( 1 ... N
)  X.  { 1 } )  =  ( ( 1 ... N
)  X.  { 0 } )  ->  ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } ) )
2624, 25nsyl 125 . . 3  |-  ( N  e.  NN  ->  -.  ( ( 1 ... N )  X.  {
1 } )  =  ( ( 1 ... N )  X.  {
0 } ) )
2726neqned 2650 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  =/=  (
( 1 ... N
)  X.  { 0 } ) )
28 neeq1 2705 . . 3  |-  ( x  =  ( ( 1 ... N )  X. 
{ 1 } )  ->  ( x  =/=  y  <->  ( ( 1 ... N )  X. 
{ 1 } )  =/=  y ) )
29 neeq2 2706 . . 3  |-  ( y  =  ( ( 1 ... N )  X. 
{ 0 } )  ->  ( ( ( 1 ... N )  X.  { 1 } )  =/=  y  <->  ( (
1 ... N )  X. 
{ 1 } )  =/=  ( ( 1 ... N )  X. 
{ 0 } ) ) )
3028, 29rspc2ev 3149 . 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 )
314, 8, 27, 30syl3anc 1292 1  |-  ( N  e.  NN  ->  E. x  e.  ( EE `  N
) E. y  e.  ( EE `  N
) x  =/=  y
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   (/)c0 3722   {csn 3959    X. cxp 4837   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558   NNcn 10631   ZZ>=cuz 11182   ...cfz 11810   EEcee 24997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-z 10962  df-uz 11183  df-fz 11811  df-ee 25000
This theorem is referenced by:  btwndiff  30865
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