Theorem iccid 9701

Description: A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)

Assertion

Ref	Expression
iccid	|- ( A e. RR* -> ( A [,] A ) = { A } )

Proof of Theorem iccid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elicc1 9700	. . . 4 |- ( ( A e. RR* /\ A e. RR* ) -> ( x e. ( A [,] A ) <-> ( x e. RR* /\ A <_ x /\ x <_ A ) ) )
2	1	anidms 394	. . 3 |- ( A e. RR* -> ( x e. ( A [,] A ) <-> ( x e. RR* /\ A <_ x /\ x <_ A ) ) )
3		xrlenlt 7822	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* ) -> ( A <_ x <-> -. x < A ) )
4		xrlenlt 7822	. . . . . . . . . . 11 |- ( ( x e. RR* /\ A e. RR* ) -> ( x <_ A <-> -. A < x ) )
5	4	ancoms 266	. . . . . . . . . 10 |- ( ( A e. RR* /\ x e. RR* ) -> ( x <_ A <-> -. A < x ) )
6		xrlttri3 9576	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ A e. RR* ) -> ( x = A <-> ( -. x < A /\ -. A < x ) ) )
7	6	biimprd 157	. . . . . . . . . . . 12 |- ( ( x e. RR* /\ A e. RR* ) -> ( ( -. x < A /\ -. A < x ) -> x = A ) )
8	7	ancoms 266	. . . . . . . . . . 11 |- ( ( A e. RR* /\ x e. RR* ) -> ( ( -. x < A /\ -. A < x ) -> x = A ) )
9	8	expcomd 1417	. . . . . . . . . 10 |- ( ( A e. RR* /\ x e. RR* ) -> ( -. A < x -> ( -. x < A -> x = A ) ) )
10	5, 9	sylbid 149	. . . . . . . . 9 |- ( ( A e. RR* /\ x e. RR* ) -> ( x <_ A -> ( -. x < A -> x = A ) ) )
11	10	com23 78	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* ) -> ( -. x < A -> ( x <_ A -> x = A ) ) )
12	3, 11	sylbid 149	. . . . . . 7 |- ( ( A e. RR* /\ x e. RR* ) -> ( A <_ x -> ( x <_ A -> x = A ) ) )
13	12	ex 114	. . . . . 6 |- ( A e. RR* -> ( x e. RR* -> ( A <_ x -> ( x <_ A -> x = A ) ) ) )
14	13	3impd 1199	. . . . 5 |- ( A e. RR* -> ( ( x e. RR* /\ A <_ x /\ x <_ A ) -> x = A ) )
15		eleq1a 2209	. . . . . 6 |- ( A e. RR* -> ( x = A -> x e. RR* ) )
16		xrleid 9579	. . . . . . 7 |- ( A e. RR* -> A <_ A )
17		breq2 3928	. . . . . . 7 |- ( x = A -> ( A <_ x <-> A <_ A ) )
18	16, 17	syl5ibrcom 156	. . . . . 6 |- ( A e. RR* -> ( x = A -> A <_ x ) )
19		breq1 3927	. . . . . . 7 |- ( x = A -> ( x <_ A <-> A <_ A ) )
20	16, 19	syl5ibrcom 156	. . . . . 6 |- ( A e. RR* -> ( x = A -> x <_ A ) )
21	15, 18, 20	3jcad 1162	. . . . 5 |- ( A e. RR* -> ( x = A -> ( x e. RR* /\ A <_ x /\ x <_ A ) ) )
22	14, 21	impbid 128	. . . 4 |- ( A e. RR* -> ( ( x e. RR* /\ A <_ x /\ x <_ A ) <-> x = A ) )
23		velsn 3539	. . . 4 |- ( x e. { A } <-> x = A )
24	22, 23	syl6bbr 197	. . 3 |- ( A e. RR* -> ( ( x e. RR* /\ A <_ x /\ x <_ A ) <-> x e. { A } ) )
25	2, 24	bitrd 187	. 2 |- ( A e. RR* -> ( x e. ( A [,] A ) <-> x e. { A } ) )
26	25	eqrdv 2135	1 |- ( A e. RR* -> ( A [,] A ) = { A } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   {csn 3522   class class class wbr 3924  (class class class)co 5767   RR*cxr 7792   < clt 7793   <_ cle 7794   [,]cicc 9667

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725  ax-pre-apti 7728

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-icc 9671

This theorem is referenced by: (None)