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Theorem elzs12 28436
Description: Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025.)
Assertion
Ref Expression
elzs12 (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦)))
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elzs12
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 3499 . 2 (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V)
2 id 22 . . . . 5 (𝐴 = (𝑥 /su (2ss𝑦)) → 𝐴 = (𝑥 /su (2ss𝑦)))
3 ovex 7464 . . . . 5 (𝑥 /su (2ss𝑦)) ∈ V
42, 3eqeltrdi 2847 . . . 4 (𝐴 = (𝑥 /su (2ss𝑦)) → 𝐴 ∈ V)
54a1i 11 . . 3 ((𝑥 ∈ ℤs𝑦 ∈ ℕ0s) → (𝐴 = (𝑥 /su (2ss𝑦)) → 𝐴 ∈ V))
65rexlimivv 3199 . 2 (∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦)) → 𝐴 ∈ V)
7 eqeq1 2739 . . . 4 (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2ss𝑦)) ↔ 𝐴 = (𝑥 /su (2ss𝑦))))
872rexbidv 3220 . . 3 (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2ss𝑦)) ↔ ∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦))))
9 df-zs12 28414 . . 3 s[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2ss𝑦))}
108, 9elab2g 3683 . 2 (𝐴 ∈ V → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦))))
111, 6, 10pm5.21nii 378 1 (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  (class class class)co 7431   /su cdivs 28228  0scnn0s 28333  sczs 28379  2sc2s 28409  scexps 28411  s[1/2]czs12 28413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-fv 6571  df-ov 7434  df-zs12 28414
This theorem is referenced by:  zzs12  28438  zs12bday  28439
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