Theorem squeeze0 9195

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 8366	. . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2	1	3ad2ant1 1045	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → ¬ 𝐴 < 𝐴)
3		breq2 4118	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴))
4		breq2 4118	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 < 𝑥 ↔ 𝐴 < 𝐴))
5	3, 4	imbi12d 234	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((0 < 𝑥 → 𝐴 < 𝑥) ↔ (0 < 𝐴 → 𝐴 < 𝐴)))
6	5	rspcva 2921	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (0 < 𝐴 → 𝐴 < 𝐴))
7	6	3adant2 1043	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (0 < 𝐴 → 𝐴 < 𝐴))
8	2, 7	mtod 669	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → ¬ 0 < 𝐴)
9		simp1 1024	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 ∈ ℝ)
10		0red 8291	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 0 ∈ ℝ)
11	9, 10	lenltd 8407	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴))
12	8, 11	mpbird 167	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 ≤ 0)
13		simp2 1025	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 0 ≤ 𝐴)
14	9, 10	letri3d 8405	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (𝐴 = 0 ↔ (𝐴 ≤ 0 ∧ 0 ≤ 𝐴)))
15	12, 13, 14	mpbir2and 953	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522   class class class wbr 4114  ℝcr 8142  0cc0 8143   < clt 8324   ≤ cle 8325

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-apti 8258

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330

This theorem is referenced by: (None)