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Theorem eltskg 10745
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 4009 . . . . 5 (𝑦 = 𝑇 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑧𝑇))
2 rexeq 3322 . . . . 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 3334 . . 3 (𝑦 = 𝑇 → (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
5 pweq 4617 . . . 4 (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6 breq2 5153 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
7 eleq2 2823 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
86, 7orbi12d 918 . . . 4 (𝑦 = 𝑇 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑇𝑧𝑇)))
95, 8raleqbidv 3343 . . 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 10744 . 2 Tarski = {𝑦 ∣ (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))}
1210, 11elab2g 3671 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 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3062  wrex 3071  wss 3949  𝒫 cpw 4603   class class class wbr 5149  cen 8936  Tarskictsk 10743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-tsk 10744
This theorem is referenced by:  eltsk2g  10746  tskpwss  10747  tsken  10749  grothtsk  10830
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