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Theorem elzdif0 30333
Description: Lemma for qqhval2 30335. (Contributed by Thierry Arnoux, 29-Oct-2017.)
Assertion
Ref Expression
elzdif0 (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))

Proof of Theorem elzdif0
StepHypRef Expression
1 eldifi 3875 . . 3 (𝑀 ∈ (ℤ ∖ {0}) → 𝑀 ∈ ℤ)
2 eldifn 3876 . . 3 (𝑀 ∈ (ℤ ∖ {0}) → ¬ 𝑀 ∈ {0})
3 elsng 4335 . . . . 5 (𝑀 ∈ ℤ → (𝑀 ∈ {0} ↔ 𝑀 = 0))
43notbid 307 . . . 4 (𝑀 ∈ ℤ → (¬ 𝑀 ∈ {0} ↔ ¬ 𝑀 = 0))
54biimpa 502 . . 3 ((𝑀 ∈ ℤ ∧ ¬ 𝑀 ∈ {0}) → ¬ 𝑀 = 0)
61, 2, 5syl2anc 696 . 2 (𝑀 ∈ (ℤ ∖ {0}) → ¬ 𝑀 = 0)
7 elz 11571 . . . . 5 (𝑀 ∈ ℤ ↔ (𝑀 ∈ ℝ ∧ (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)))
81, 7sylib 208 . . . 4 (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℝ ∧ (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)))
98simprd 482 . . 3 (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))
10 3orass 1075 . . 3 ((𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ) ↔ (𝑀 = 0 ∨ (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)))
119, 10sylib 208 . 2 (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 = 0 ∨ (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)))
12 orel1 396 . 2 𝑀 = 0 → ((𝑀 = 0 ∨ (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)))
136, 11, 12sylc 65 1 (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3o 1071   = wceq 1632  wcel 2139  cdif 3712  {csn 4321  cr 10127  0cc0 10128  -cneg 10459  cn 11212  cz 11569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-neg 10461  df-z 11570
This theorem is referenced by: (None)
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