Theorem squeeze0 7651

Description: If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Assertion

Ref	Expression
squeeze0	⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → A = 0)

Distinct variable group:   x,A

Proof of Theorem squeeze0

Step	Hyp	Ref	Expression
1		ltnr 6892	. . . . 5 ⊢ (A ∈ ℝ → ¬ A < A)
2	1	3ad2ant1 924	. . . 4 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → ¬ A < A)
3		breq2 3759	. . . . . . 7 ⊢ (x = A → (0 < x ↔ 0 < A))
4		breq2 3759	. . . . . . 7 ⊢ (x = A → (A < x ↔ A < A))
5	3, 4	imbi12d 223	. . . . . 6 ⊢ (x = A → ((0 < x → A < x) ↔ (0 < A → A < A)))
6	5	rspcva 2648	. . . . 5 ⊢ ((A ∈ ℝ ∧ ∀x ∈ ℝ (0 < x → A < x)) → (0 < A → A < A))
7	6	3adant2 922	. . . 4 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → (0 < A → A < A))
8	2, 7	mtod 588	. . 3 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → ¬ 0 < A)
9		simp1 903	. . . 4 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → A ∈ ℝ)
10		0red 6826	. . . 4 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → 0 ∈ ℝ)
11	9, 10	lenltd 6931	. . 3 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → (A ≤ 0 ↔ ¬ 0 < A))
12	8, 11	mpbird 156	. 2 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → A ≤ 0)
13		simp2 904	. 2 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → 0 ≤ A)
14	9, 10	letri3d 6930	. 2 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → (A = 0 ↔ (A ≤ 0 ∧ 0 ≤ A)))
15	12, 13, 14	mpbir2and 850	1 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → A = 0)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300   class class class wbr 3755  ℝcr 6710  0cc0 6711   < clt 6857   ≤ cle 6858

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-bndl 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-1re 6777  ax-addrcl 6780  ax-rnegex 6792  ax-pre-ltirr 6795  ax-pre-apti 6798

This theorem depends on definitions:  df-bi 110  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-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-xp 4294  df-cnv 4296  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863

This theorem is referenced by: (None)