Theorem elzs12 28376

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 3455	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V)
2		id 22	. . . . 5 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 = (𝑥 /su (2s↑s𝑦)))
3		ovex 7374	. . . . 5 ⊢ (𝑥 /su (2s↑s𝑦)) ∈ V
4	2, 3	eqeltrdi 2837	. . . 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 3172	. 2 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
7		eqeq1 2734	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ 𝐴 = (𝑥 /su (2s↑s𝑦))))
8	7	2rexbidv 3195	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
9		df-zs12 28331	. . 3 ⊢ ℤs[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦))}
10	8, 9	elab2g 3634	. 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 1541   ∈ wcel 2110  ∃wrex 3054  Vcvv 3434  (class class class)co 7341   /_su cdivs 28119  ℕ_0scnn0s 28235  ℤ_sczs 28295  2_sc2s 28326  ↑_scexps 28328  ℤ_s[1/2]czs12 28330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rex 3055  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-sn 4575  df-pr 4577  df-uni 4858  df-iota 6433  df-fv 6485  df-ov 7344  df-zs12 28331

This theorem is referenced by:  zzs12  28378  zs12no  28379  zs12addscl  28380  zs12negscl  28381  zs12half  28383  zs12zodd  28385  zs12ge0  28386  zs12bday  28387