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Theorem elzs12 28422
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 3500 . 2 (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V)
2 id 22 . . . . 5 (𝐴 = (𝑥 /su (2ss𝑦)) → 𝐴 = (𝑥 /su (2ss𝑦)))
3 ovex 7465 . . . . 5 (𝑥 /su (2ss𝑦)) ∈ V
42, 3eqeltrdi 2848 . . . 4 (𝐴 = (𝑥 /su (2ss𝑦)) → 𝐴 ∈ V)
54a1i 11 . . 3 ((𝑥 ∈ ℤs𝑦 ∈ ℕ0s) → (𝐴 = (𝑥 /su (2ss𝑦)) → 𝐴 ∈ V))
65rexlimivv 3200 . 2 (∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦)) → 𝐴 ∈ V)
7 eqeq1 2740 . . . 4 (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2ss𝑦)) ↔ 𝐴 = (𝑥 /su (2ss𝑦))))
872rexbidv 3221 . . 3 (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2ss𝑦)) ↔ ∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2ss𝑦))))
9 df-zs12 28400 . . 3 s[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2ss𝑦))}
108, 9elab2g 3679 . 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 1539  wcel 2107  wrex 3069  Vcvv 3479  (class class class)co 7432   /su cdivs 28214  0scnn0s 28319  sczs 28365  2sc2s 28395  scexps 28397  s[1/2]czs12 28399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rex 3070  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513  df-fv 6568  df-ov 7435  df-zs12 28400
This theorem is referenced by:  zzs12  28424  zs12bday  28425
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