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Theorem eltskg 10788
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 4022 . . . . 5 (𝑦 = 𝑇 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑧𝑇))
2 rexeq 3320 . . . . 5 (𝑦 = 𝑇 → (∃𝑤𝑦 𝒫 𝑧𝑤 ↔ ∃𝑤𝑇 𝒫 𝑧𝑤))
31, 2anbi12d 632 . . . 4 (𝑦 = 𝑇 → ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
43raleqbi1dv 3336 . . 3 (𝑦 = 𝑇 → (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
5 pweq 4619 . . . 4 (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6 breq2 5152 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
7 eleq2 2828 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
86, 7orbi12d 918 . . . 4 (𝑦 = 𝑇 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑇𝑧𝑇)))
95, 8raleqbidv 3344 . . 3 (𝑦 = 𝑇 → (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)))
104, 9anbi12d 632 . 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 10787 . 2 Tarski = {𝑦 ∣ (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))}
1210, 11elab2g 3683 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cen 8981  Tarskictsk 10786
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-tsk 10787
This theorem is referenced by:  eltsk2g  10789  tskpwss  10790  tsken  10792  grothtsk  10873
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