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Mirrors > Home > MPE Home > Th. List > elzs12 | Structured version Visualization version GIF version |
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 3499 | . 2 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 = (𝑥 /su (2s↑s𝑦))) | |
3 | ovex 7464 | . . . . 5 ⊢ (𝑥 /su (2s↑s𝑦)) ∈ V | |
4 | 2, 3 | eqeltrdi 2847 | . . . 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 3199 | . 2 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V) |
7 | eqeq1 2739 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ 𝐴 = (𝑥 /su (2s↑s𝑦)))) | |
8 | 7 | 2rexbidv 3220 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))) |
9 | df-zs12 28414 | . . 3 ⊢ ℤs[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦))} | |
10 | 8, 9 | elab2g 3683 | . 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 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 (class class class)co 7431 /su cdivs 28228 ℕ0scnn0s 28333 ℤsczs 28379 2sc2s 28409 ↑scexps 28411 ℤs[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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