Theorem squeeze0 9083

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 8255	. . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2	1	3ad2ant1 1044	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → ¬ 𝐴 < 𝐴)
3		breq2 4092	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴))
4		breq2 4092	. . . . . . 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 2908	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (0 < 𝐴 → 𝐴 < 𝐴))
7	6	3adant2 1042	. . . 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 1023	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 ∈ ℝ)
10		0red 8179	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 0 ∈ ℝ)
11	9, 10	lenltd 8296	. . 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 1024	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 0 ≤ 𝐴)
14	9, 10	letri3d 8294	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (𝐴 = 0 ↔ (𝐴 ≤ 0 ∧ 0 ≤ 𝐴)))
15	12, 13, 14	mpbir2and 952	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510   class class class wbr 4088  ℝcr 8030  0cc0 8031   < clt 8213   ≤ cle 8214

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-apti 8146

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219

This theorem is referenced by: (None)