Theorem iccid 9708

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 9707	. . . 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 7829	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* ) -> ( A <_ x <-> -. x < A ) )
4		xrlenlt 7829	. . . . . . . . . . 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 9583	. . . . . . . . . . . . 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 2211	. . . . . 6 |- ( A e. RR* -> ( x = A -> x e. RR* ) )
16		xrleid 9586	. . . . . . 7 |- ( A e. RR* -> A <_ A )
17		breq2 3933	. . . . . . 7 |- ( x = A -> ( A <_ x <-> A <_ A ) )
18	16, 17	syl5ibrcom 156	. . . . . 6 |- ( A e. RR* -> ( x = A -> A <_ x ) )
19		breq1 3932	. . . . . . 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 3544	. . . 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 2137	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 3527   class class class wbr 3929  (class class class)co 5774   RR*cxr 7799   < clt 7800   <_ cle 7801   [,]cicc 9674

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-apti 7735

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-icc 9678

This theorem is referenced by: (None)