Theorem iccid 8564

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 ∈ ℝ* → (A[,]A) = {A})

Proof of Theorem iccid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elicc1 8563	. . . 4 ⊢ ((A ∈ ℝ* ∧ A ∈ ℝ*) → (x ∈ (A[,]A) ↔ (x ∈ ℝ* ∧ A ≤ x ∧ x ≤ A)))
2	1	anidms 377	. . 3 ⊢ (A ∈ ℝ* → (x ∈ (A[,]A) ↔ (x ∈ ℝ* ∧ A ≤ x ∧ x ≤ A)))
3		xrlenlt 6881	. . . . . . . 8 ⊢ ((A ∈ ℝ* ∧ x ∈ ℝ*) → (A ≤ x ↔ ¬ x < A))
4		xrlenlt 6881	. . . . . . . . . . 11 ⊢ ((x ∈ ℝ* ∧ A ∈ ℝ*) → (x ≤ A ↔ ¬ A < x))
5	4	ancoms 255	. . . . . . . . . 10 ⊢ ((A ∈ ℝ* ∧ x ∈ ℝ*) → (x ≤ A ↔ ¬ A < x))
6		xrlttri3 8488	. . . . . . . . . . . . 13 ⊢ ((x ∈ ℝ* ∧ A ∈ ℝ*) → (x = A ↔ (¬ x < A ∧ ¬ A < x)))
7	6	biimprd 147	. . . . . . . . . . . 12 ⊢ ((x ∈ ℝ* ∧ A ∈ ℝ*) → ((¬ x < A ∧ ¬ A < x) → x = A))
8	7	ancoms 255	. . . . . . . . . . 11 ⊢ ((A ∈ ℝ* ∧ x ∈ ℝ*) → ((¬ x < A ∧ ¬ A < x) → x = A))
9	8	expcomd 1327	. . . . . . . . . 10 ⊢ ((A ∈ ℝ* ∧ x ∈ ℝ*) → (¬ A < x → (¬ x < A → x = A)))
10	5, 9	sylbid 139	. . . . . . . . 9 ⊢ ((A ∈ ℝ* ∧ x ∈ ℝ*) → (x ≤ A → (¬ x < A → x = A)))
11	10	com23 72	. . . . . . . 8 ⊢ ((A ∈ ℝ* ∧ x ∈ ℝ*) → (¬ x < A → (x ≤ A → x = A)))
12	3, 11	sylbid 139	. . . . . . 7 ⊢ ((A ∈ ℝ* ∧ x ∈ ℝ*) → (A ≤ x → (x ≤ A → x = A)))
13	12	ex 108	. . . . . 6 ⊢ (A ∈ ℝ* → (x ∈ ℝ* → (A ≤ x → (x ≤ A → x = A))))
14	13	3impd 1117	. . . . 5 ⊢ (A ∈ ℝ* → ((x ∈ ℝ* ∧ A ≤ x ∧ x ≤ A) → x = A))
15		eleq1a 2106	. . . . . 6 ⊢ (A ∈ ℝ* → (x = A → x ∈ ℝ*))
16		xrleid 8490	. . . . . . 7 ⊢ (A ∈ ℝ* → A ≤ A)
17		breq2 3759	. . . . . . 7 ⊢ (x = A → (A ≤ x ↔ A ≤ A))
18	16, 17	syl5ibrcom 146	. . . . . 6 ⊢ (A ∈ ℝ* → (x = A → A ≤ x))
19		breq1 3758	. . . . . . 7 ⊢ (x = A → (x ≤ A ↔ A ≤ A))
20	16, 19	syl5ibrcom 146	. . . . . 6 ⊢ (A ∈ ℝ* → (x = A → x ≤ A))
21	15, 18, 20	3jcad 1084	. . . . 5 ⊢ (A ∈ ℝ* → (x = A → (x ∈ ℝ* ∧ A ≤ x ∧ x ≤ A)))
22	14, 21	impbid 120	. . . 4 ⊢ (A ∈ ℝ* → ((x ∈ ℝ* ∧ A ≤ x ∧ x ≤ A) ↔ x = A))
23		elsn 3382	. . . 4 ⊢ (x ∈ {A} ↔ x = A)
24	22, 23	syl6bbr 187	. . 3 ⊢ (A ∈ ℝ* → ((x ∈ ℝ* ∧ A ≤ x ∧ x ≤ A) ↔ x ∈ {A}))
25	2, 24	bitrd 177	. 2 ⊢ (A ∈ ℝ* → (x ∈ (A[,]A) ↔ x ∈ {A}))
26	25	eqrdv 2035	1 ⊢ (A ∈ ℝ* → (A[,]A) = {A})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  {csn 3367   class class class wbr 3755  (class class class)co 5455  ℝ^*cxr 6856   < clt 6857   ≤ cle 6858  [,]cicc 8530

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-ltirr 6795  ax-pre-apti 6798

This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-icc 8534

This theorem is referenced by: (None)