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Theorem iccen 10795
Description: Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
Assertion
Ref Expression
iccen  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
~~  ( A [,] B ) )

Proof of Theorem iccen
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A ) ) )  =  ( x  e.  ( 0 [,] 1
)  |->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A
) ) )
21iccf1o 10794 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( x  e.  ( 0 [,] 1 ) 
|->  ( ( x  x.  B )  +  ( ( 1  -  x
)  x.  A ) ) ) : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B )  /\  `' ( x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A ) ) )  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) ) )
32simpld 445 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B
)  +  ( ( 1  -  x )  x.  A ) ) ) : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) )
4 ovex 5899 . . 3  |-  ( 0 [,] 1 )  e. 
_V
5 ovex 5899 . . 3  |-  ( A [,] B )  e. 
_V
6 f1oen2g 6894 . . 3  |-  ( ( ( 0 [,] 1
)  e.  _V  /\  ( A [,] B )  e.  _V  /\  (
x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B
)  +  ( ( 1  -  x )  x.  A ) ) ) : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) )  ->  ( 0 [,] 1 )  ~~  ( A [,] B ) )
74, 5, 6mp3an12 1267 . 2  |-  ( ( x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B
)  +  ( ( 1  -  x )  x.  A ) ) ) : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B )  -> 
( 0 [,] 1
)  ~~  ( A [,] B ) )
83, 7syl 15 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
0 [,] 1 ) 
~~  ( A [,] B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   -1-1-onto->wf1o 5270  (class class class)co 5874    ~~ cen 6876   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053    / cdiv 9439   [,]cicc 10675
This theorem is referenced by:  opnreen  18352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-rp 10371  df-icc 10679
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