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Theorem eltskg 10741
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 4007 . . . . 5 (𝑦 = 𝑇 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑧𝑇))
2 rexeq 3321 . . . . 5 (𝑦 = 𝑇 → (∃𝑤𝑦 𝒫 𝑧𝑤 ↔ ∃𝑤𝑇 𝒫 𝑧𝑤))
31, 2anbi12d 631 . . . 4 (𝑦 = 𝑇 → ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
43raleqbi1dv 3333 . . 3 (𝑦 = 𝑇 → (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
5 pweq 4615 . . . 4 (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6 breq2 5151 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
7 eleq2 2822 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
86, 7orbi12d 917 . . . 4 (𝑦 = 𝑇 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑇𝑧𝑇)))
95, 8raleqbidv 3342 . . 3 (𝑦 = 𝑇 → (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)))
104, 9anbi12d 631 . 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 10740 . 2 Tarski = {𝑦 ∣ (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))}
1210, 11elab2g 3669 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 396  wo 845   = wceq 1541  wcel 2106  wral 3061  wrex 3070  wss 3947  𝒫 cpw 4601   class class class wbr 5147  cen 8932  Tarskictsk 10739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-tsk 10740
This theorem is referenced by:  eltsk2g  10742  tskpwss  10743  tsken  10745  grothtsk  10826
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