Theorem iccid 9882

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 9881	. . . 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 7984	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* ) -> ( A <_ x <-> -. x < A ) )
4		xrlenlt 7984	. . . . . . . . . . 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 9754	. . . . . . . . . . . . 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 1434	. . . . . . . . . 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 1216	. . . . 5 |- ( A e. RR* -> ( ( x e. RR* /\ A <_ x /\ x <_ A ) -> x = A ) )
15		eleq1a 2242	. . . . . 6 |- ( A e. RR* -> ( x = A -> x e. RR* ) )
16		xrleid 9757	. . . . . . 7 |- ( A e. RR* -> A <_ A )
17		breq2 3993	. . . . . . 7 |- ( x = A -> ( A <_ x <-> A <_ A ) )
18	16, 17	syl5ibrcom 156	. . . . . 6 |- ( A e. RR* -> ( x = A -> A <_ x ) )
19		breq1 3992	. . . . . . 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 1173	. . . . 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 3600	. . . 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 2168	1 |- ( A e. RR* -> ( A [,] A ) = { A } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   {csn 3583   class class class wbr 3989  (class class class)co 5853   RR*cxr 7953   < clt 7954   <_ cle 7955   [,]cicc 9848

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-apti 7889

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-icc 9852

This theorem is referenced by: (None)