Theorem iccid 9861

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 9860	. . . 4 |- ( ( A e. RR* /\ A e. RR* ) -> ( x e. ( A [,] A ) <-> ( x e. RR* /\ A <_ x /\ x <_ A ) ) )
2	1	anidms 395	. . 3 |- ( A e. RR* -> ( x e. ( A [,] A ) <-> ( x e. RR* /\ A <_ x /\ x <_ A ) ) )
3		xrlenlt 7963	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* ) -> ( A <_ x <-> -. x < A ) )
4		xrlenlt 7963	. . . . . . . . . . 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 9733	. . . . . . . . . . . . 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 1429	. . . . . . . . . 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 1211	. . . . 5 |- ( A e. RR* -> ( ( x e. RR* /\ A <_ x /\ x <_ A ) -> x = A ) )
15		eleq1a 2238	. . . . . 6 |- ( A e. RR* -> ( x = A -> x e. RR* ) )
16		xrleid 9736	. . . . . . 7 |- ( A e. RR* -> A <_ A )
17		breq2 3986	. . . . . . 7 |- ( x = A -> ( A <_ x <-> A <_ A ) )
18	16, 17	syl5ibrcom 156	. . . . . 6 |- ( A e. RR* -> ( x = A -> A <_ x ) )
19		breq1 3985	. . . . . . 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 1168	. . . . 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 3593	. . . 4 |- ( x e. { A } <-> x = A )
24	22, 23	bitr4di 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 2163	1 |- ( A e. RR* -> ( A [,] A ) = { A } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   {csn 3576   class class class wbr 3982  (class class class)co 5842   RR*cxr 7932   < clt 7933   <_ cle 7934   [,]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-pre-ltirr 7865  ax-pre-apti 7868

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-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-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-icc 9831

This theorem is referenced by: (None)