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Theorem eltskg 9451
 Description: Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
eltskg (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
Distinct variable group:   𝑤,𝑇,𝑧
Allowed substitution hints:   𝑉(𝑧,𝑤)

Proof of Theorem eltskg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sseq2 3590 . . . . 5 (𝑦 = 𝑇 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑧𝑇))
2 rexeq 3116 . . . . 5 (𝑦 = 𝑇 → (∃𝑤𝑦 𝒫 𝑧𝑤 ↔ ∃𝑤𝑇 𝒫 𝑧𝑤))
31, 2anbi12d 743 . . . 4 (𝑦 = 𝑇 → ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
43raleqbi1dv 3123 . . 3 (𝑦 = 𝑇 → (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
5 pweq 4111 . . . 4 (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6 breq2 4587 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
7 eleq2 2677 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
86, 7orbi12d 742 . . . 4 (𝑦 = 𝑇 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑇𝑧𝑇)))
95, 8raleqbidv 3129 . . 3 (𝑦 = 𝑇 → (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)))
104, 9anbi12d 743 . 2 (𝑦 = 𝑇 → ((∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
11 df-tsk 9450 . 2 Tarski = {𝑦 ∣ (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))}
1210, 11elab2g 3322 1 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ≈ cen 7838  Tarskictsk 9449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-tsk 9450 This theorem is referenced by:  eltsk2g  9452  tskpwss  9453  tsken  9455  grothtsk  9536
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