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| Description: Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025.) | 
| Ref | Expression | 
|---|---|
| elzs12 | ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elex 3500 | . 2 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V) | |
| 2 | id 22 | . . . . 5 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 = (𝑥 /su (2s↑s𝑦))) | |
| 3 | ovex 7465 | . . . . 5 ⊢ (𝑥 /su (2s↑s𝑦)) ∈ V | |
| 4 | 2, 3 | eqeltrdi 2848 | . . . 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 3200 | . 2 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V) | 
| 7 | eqeq1 2740 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ 𝐴 = (𝑥 /su (2s↑s𝑦)))) | |
| 8 | 7 | 2rexbidv 3221 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))) | 
| 9 | df-zs12 28400 | . . 3 ⊢ ℤs[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦))} | |
| 10 | 8, 9 | elab2g 3679 | . 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))) | 
| 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 ℕ0scnn0s 28319 ℤsczs 28365 2sc2s 28395 ↑scexps 28397 ℤs[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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