Theorem iccen 9942

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 7887	. . 3 |- RR e. _V
2		unitssre 9941	. . 3 |- ( 0 [,] 1 ) C_ RR
3	1, 2	ssexi 4120	. 2 |- ( 0 [,] 1 ) e. _V
4		iccssre 9891	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
5		ssexg 4121	. . . 4 |- ( ( ( A [,] B ) C_ RR /\ RR e. _V ) -> ( A [,] B ) e. _V )
6	4, 1, 5	sylancl 410	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) e. _V )
7	6	3adant3 1007	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A [,] B ) e. _V )
8		eqid 2165	. . . 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 9940	. . 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 111	. 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 6721	. 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 1332	1 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( 0 [,] 1 ) ~~ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   class class class wbr 3982   |-> cmpt 4043   `'ccnv 4603   -1-1-onto->wf1o 5187  (class class class)co 5842   ~~ cen 6704   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   - cmin 8069   / cdiv 8568   [,]cicc 9827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-en 6707  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-rp 9590  df-icc 9831

This theorem is referenced by: (None)