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Theorem squeeze0 8259
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 < 𝑥𝐴 < 𝑥)) → 𝐴 = 0)
Distinct variable group:   𝑥,𝐴

Proof of Theorem squeeze0
StepHypRef Expression
1 ltnr 7465 . . . . 5 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
213ad2ant1 960 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → ¬ 𝐴 < 𝐴)
3 breq2 3815 . . . . . . 7 (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴))
4 breq2 3815 . . . . . . 7 (𝑥 = 𝐴 → (𝐴 < 𝑥𝐴 < 𝐴))
53, 4imbi12d 232 . . . . . 6 (𝑥 = 𝐴 → ((0 < 𝑥𝐴 < 𝑥) ↔ (0 < 𝐴𝐴 < 𝐴)))
65rspcva 2710 . . . . 5 ((𝐴 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → (0 < 𝐴𝐴 < 𝐴))
763adant2 958 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → (0 < 𝐴𝐴 < 𝐴))
82, 7mtod 622 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → ¬ 0 < 𝐴)
9 simp1 939 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → 𝐴 ∈ ℝ)
10 0red 7392 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → 0 ∈ ℝ)
119, 10lenltd 7504 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴))
128, 11mpbird 165 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → 𝐴 ≤ 0)
13 simp2 940 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → 0 ≤ 𝐴)
149, 10letri3d 7503 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → (𝐴 = 0 ↔ (𝐴 ≤ 0 ∧ 0 ≤ 𝐴)))
1512, 13, 14mpbir2and 886 1 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → 𝐴 = 0)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  w3a 920   = wceq 1285  wcel 1434  wral 2353   class class class wbr 3811  cr 7252  0cc0 7253   < clt 7425  cle 7426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1re 7342  ax-addrcl 7345  ax-rnegex 7357  ax-pre-ltirr 7360  ax-pre-apti 7363
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-xp 4407  df-cnv 4409  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431
This theorem is referenced by: (None)
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