Theorem elzs12 28368

Description: Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025.)

Assertion

Ref	Expression
elzs12	⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elzs12

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3459	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V)
2		id 22	. . . . 5 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 = (𝑥 /su (2s↑s𝑦)))
3		ovex 7386	. . . . 5 ⊢ (𝑥 /su (2s↑s𝑦)) ∈ V
4	2, 3	eqeltrdi 2836	. . . 4 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
5	4	a1i 11	. . 3 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℕ0s) → (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V))
6	5	rexlimivv 3171	. 2 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
7		eqeq1 2733	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ 𝐴 = (𝑥 /su (2s↑s𝑦))))
8	7	2rexbidv 3194	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
9		df-zs12 28325	. . 3 ⊢ ℤs[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦))}
10	8, 9	elab2g 3638	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
11	1, 6, 10	pm5.21nii 378	1 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3438  (class class class)co 7353   /_su cdivs 28113  ℕ_0scnn0s 28229  ℤ_sczs 28289  2_sc2s 28320  ↑_scexps 28322  ℤ_s[1/2]czs12 28324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rex 3054  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-sn 4580  df-pr 4582  df-uni 4862  df-iota 6442  df-fv 6494  df-ov 7356  df-zs12 28325

This theorem is referenced by:  zzs12  28370  zs12no  28371  zs12addscl  28372  zs12negscl  28373  zs12half  28375  zs12zodd  28377  zs12ge0  28378  zs12bday  28379