Theorem iccid 9994

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 9993	. . . 4 |- ( ( A e. RR* /\ A e. RR* ) -> ( x e. ( A [,] A ) <-> ( x e. RR* /\ A <_ x /\ x <_ A ) ) )
2	1	anidms 397	. . 3 |- ( A e. RR* -> ( x e. ( A [,] A ) <-> ( x e. RR* /\ A <_ x /\ x <_ A ) ) )
3		xrlenlt 8086	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* ) -> ( A <_ x <-> -. x < A ) )
4		xrlenlt 8086	. . . . . . . . . . 11 |- ( ( x e. RR* /\ A e. RR* ) -> ( x <_ A <-> -. A < x ) )
5	4	ancoms 268	. . . . . . . . . 10 |- ( ( A e. RR* /\ x e. RR* ) -> ( x <_ A <-> -. A < x ) )
6		xrlttri3 9866	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ A e. RR* ) -> ( x = A <-> ( -. x < A /\ -. A < x ) ) )
7	6	biimprd 158	. . . . . . . . . . . 12 |- ( ( x e. RR* /\ A e. RR* ) -> ( ( -. x < A /\ -. A < x ) -> x = A ) )
8	7	ancoms 268	. . . . . . . . . . 11 |- ( ( A e. RR* /\ x e. RR* ) -> ( ( -. x < A /\ -. A < x ) -> x = A ) )
9	8	expcomd 1452	. . . . . . . . . 10 |- ( ( A e. RR* /\ x e. RR* ) -> ( -. A < x -> ( -. x < A -> x = A ) ) )
10	5, 9	sylbid 150	. . . . . . . . 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 150	. . . . . . 7 |- ( ( A e. RR* /\ x e. RR* ) -> ( A <_ x -> ( x <_ A -> x = A ) ) )
13	12	ex 115	. . . . . 6 |- ( A e. RR* -> ( x e. RR* -> ( A <_ x -> ( x <_ A -> x = A ) ) ) )
14	13	3impd 1223	. . . . 5 |- ( A e. RR* -> ( ( x e. RR* /\ A <_ x /\ x <_ A ) -> x = A ) )
15		eleq1a 2265	. . . . . 6 |- ( A e. RR* -> ( x = A -> x e. RR* ) )
16		xrleid 9869	. . . . . . 7 |- ( A e. RR* -> A <_ A )
17		breq2 4034	. . . . . . 7 |- ( x = A -> ( A <_ x <-> A <_ A ) )
18	16, 17	syl5ibrcom 157	. . . . . 6 |- ( A e. RR* -> ( x = A -> A <_ x ) )
19		breq1 4033	. . . . . . 7 |- ( x = A -> ( x <_ A <-> A <_ A ) )
20	16, 19	syl5ibrcom 157	. . . . . 6 |- ( A e. RR* -> ( x = A -> x <_ A ) )
21	15, 18, 20	3jcad 1180	. . . . 5 |- ( A e. RR* -> ( x = A -> ( x e. RR* /\ A <_ x /\ x <_ A ) ) )
22	14, 21	impbid 129	. . . 4 |- ( A e. RR* -> ( ( x e. RR* /\ A <_ x /\ x <_ A ) <-> x = A ) )
23		velsn 3636	. . . 4 |- ( x e. { A } <-> x = A )
24	22, 23	bitr4di 198	. . 3 |- ( A e. RR* -> ( ( x e. RR* /\ A <_ x /\ x <_ A ) <-> x e. { A } ) )
25	2, 24	bitrd 188	. 2 |- ( A e. RR* -> ( x e. ( A [,] A ) <-> x e. { A } ) )
26	25	eqrdv 2191	1 |- ( A e. RR* -> ( A [,] A ) = { A } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   {csn 3619   class class class wbr 4030  (class class class)co 5919   RR*cxr 8055   < clt 8056   <_ cle 8057   [,]cicc 9960

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-pre-ltirr 7986  ax-pre-apti 7989

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-icc 9964

This theorem is referenced by: (None)