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Theorem eltskg 9773
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 3774 . . . . 5 (𝑦 = 𝑇 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑧𝑇))
2 rexeq 3287 . . . . 5 (𝑦 = 𝑇 → (∃𝑤𝑦 𝒫 𝑧𝑤 ↔ ∃𝑤𝑇 𝒫 𝑧𝑤))
31, 2anbi12d 608 . . . 4 (𝑦 = 𝑇 → ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
43raleqbi1dv 3294 . . 3 (𝑦 = 𝑇 → (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤)))
5 pweq 4298 . . . 4 (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6 breq2 4788 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
7 eleq2 2838 . . . . 5 (𝑦 = 𝑇 → (𝑧𝑦𝑧𝑇))
86, 7orbi12d 883 . . . 4 (𝑦 = 𝑇 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑇𝑧𝑇)))
95, 8raleqbidv 3300 . . 3 (𝑦 = 𝑇 → (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)))
104, 9anbi12d 608 . 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 9772 . 2 Tarski = {𝑦 ∣ (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))}
1210, 11elab2g 3502 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 196  wa 382  wo 826   = wceq 1630  wcel 2144  wral 3060  wrex 3061  wss 3721  𝒫 cpw 4295   class class class wbr 4784  cen 8105  Tarskictsk 9771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-tsk 9772
This theorem is referenced by:  eltsk2g  9774  tskpwss  9775  tsken  9777  grothtsk  9858
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