Theorem iccen 10388

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.

Step	Hyp	Ref	Expression
1		reex 8303	. . 3 |- RR e. _V
2		unitssre 10387	. . 3 |- ( 0 [,] 1 ) C_ RR
3	1, 2	ssexi 4266	. 2 |- ( 0 [,] 1 ) e. _V
4		iccssre 10336	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
5		ssexg 4267	. . . 4 |- ( ( ( A [,] B ) C_ RR /\ RR e. _V ) -> ( A [,] B ) e. _V )
6	4, 1, 5	sylancl 417	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) e. _V )
7	6	3adant3 1048	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A [,] B ) e. _V )
8		eqid 2238	. . . 4 |- ( x e. ( 0 [,] 1 ) |-> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) ) = ( x e. ( 0 [,] 1 ) |-> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) )
9	8	iccf1o 10386	. . 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 ) ) ) ) )
10	9	simpld 112	. 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 ) )
11		f1oen2g 7031	. 2 |- ( ( ( 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 ) )
12	3, 7, 10, 11	mp3an2i 1383	1 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( 0 [,] 1 ) ~~ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4125   |-> cmpt 4187   `'ccnv 4768   -1-1-onto->wf1o 5371  (class class class)co 6075   ~~ cen 7010   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   - cmin 8487   / cdiv 8992   [,]cicc 10272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-en 7013  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-rp 10034  df-icc 10276

This theorem is referenced by: (None)