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Theorem axlowdim1 24989
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 24990. (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 9642 . . . 4  |-  1  e.  RR
21fconst6 5773 . . 3  |-  ( ( 1 ... N )  X.  { 1 } ) : ( 1 ... N ) --> RR
3 elee 24924 . . 3  |-  ( N  e.  NN  ->  (
( ( 1 ... N )  X.  {
1 } )  e.  ( EE `  N
)  <->  ( ( 1 ... N )  X. 
{ 1 } ) : ( 1 ... N ) --> RR ) )
42, 3mpbiri 237 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  e.  ( EE `  N ) )
5 0re 9643 . . . 4  |-  0  e.  RR
65fconst6 5773 . . 3  |-  ( ( 1 ... N )  X.  { 0 } ) : ( 1 ... N ) --> RR
7 elee 24924 . . 3  |-  ( N  e.  NN  ->  (
( ( 1 ... N )  X.  {
0 } )  e.  ( EE `  N
)  <->  ( ( 1 ... N )  X. 
{ 0 } ) : ( 1 ... N ) --> RR ) )
86, 7mpbiri 237 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 0 } )  e.  ( EE `  N ) )
9 ax-1ne0 9608 . . . . . . 7  |-  1  =/=  0
109neii 2626 . . . . . 6  |-  -.  1  =  0
11 1ex 9638 . . . . . . 7  |-  1  e.  _V
1211sneqr 4139 . . . . . 6  |-  ( { 1 }  =  {
0 }  ->  1  =  0 )
1310, 12mto 180 . . . . 5  |-  -.  {
1 }  =  {
0 }
14 elnnuz 11195 . . . . . . . . 9  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
15 eluzfz1 11806 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
1614, 15sylbi 199 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
17 ne0i 3737 . . . . . . . 8  |-  ( 1  e.  ( 1 ... N )  ->  (
1 ... N )  =/=  (/) )
1816, 17syl 17 . . . . . . 7  |-  ( N  e.  NN  ->  (
1 ... N )  =/=  (/) )
19 rnxp 5267 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 1 } )  =  { 1 } )
2018, 19syl 17 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
1 } )  =  { 1 } )
21 rnxp 5267 . . . . . . 7  |-  ( ( 1 ... N )  =/=  (/)  ->  ran  ( ( 1 ... N )  X.  { 0 } )  =  { 0 } )
2218, 21syl 17 . . . . . 6  |-  ( N  e.  NN  ->  ran  ( ( 1 ... N )  X.  {
0 } )  =  { 0 } )
2320, 22eqeq12d 2466 . . . . 5  |-  ( N  e.  NN  ->  ( ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } )  <->  { 1 }  =  { 0 } ) )
2413, 23mtbiri 305 . . . 4  |-  ( N  e.  NN  ->  -.  ran  ( ( 1 ... N )  X.  {
1 } )  =  ran  ( ( 1 ... N )  X. 
{ 0 } ) )
25 rneq 5060 . . . 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 2631 . 2  |-  ( N  e.  NN  ->  (
( 1 ... N
)  X.  { 1 } )  =/=  (
( 1 ... N
)  X.  { 0 } ) )
28 neeq1 2686 . . 3  |-  ( x  =  ( ( 1 ... N )  X. 
{ 1 } )  ->  ( x  =/=  y  <->  ( ( 1 ... N )  X. 
{ 1 } )  =/=  y ) )
29 neeq2 2687 . . 3  |-  ( y  =  ( ( 1 ... N )  X. 
{ 0 } )  ->  ( ( ( 1 ... N )  X.  { 1 } )  =/=  y  <->  ( (
1 ... N )  X. 
{ 1 } )  =/=  ( ( 1 ... N )  X. 
{ 0 } ) ) )
3028, 29rspc2ev 3161 . 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 1268 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 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   (/)c0 3731   {csn 3968    X. cxp 4832   ran crn 4835   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   NNcn 10609   ZZ>=cuz 11159   ...cfz 11784   EEcee 24918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-z 10938  df-uz 11160  df-fz 11785  df-ee 24921
This theorem is referenced by:  btwndiff  30794
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