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Theorem eltsk2g 9611
Description: Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
eltsk2g (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
Distinct variable group:   𝑧,𝑇
Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem eltsk2g
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eltskg 9610 . 2 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
2 nfra1 2970 . . . . . . 7 𝑧𝑧𝑇 𝒫 𝑧𝑇
3 pweq 4194 . . . . . . . . . . . 12 (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
43sseq1d 3665 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝒫 𝑧𝑇 ↔ 𝒫 𝑤𝑇))
54rspccva 3339 . . . . . . . . . 10 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑤𝑇) → 𝒫 𝑤𝑇)
65adantlr 751 . . . . . . . . 9 (((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) ∧ 𝑤𝑇) → 𝒫 𝑤𝑇)
7 vpwex 4879 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
87elpw 4197 . . . . . . . . . 10 (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧𝑤)
9 ssel 3630 . . . . . . . . . 10 (𝒫 𝑤𝑇 → (𝒫 𝑧 ∈ 𝒫 𝑤 → 𝒫 𝑧𝑇))
108, 9syl5bir 233 . . . . . . . . 9 (𝒫 𝑤𝑇 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
116, 10syl 17 . . . . . . . 8 (((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) ∧ 𝑤𝑇) → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
1211rexlimdva 3060 . . . . . . 7 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) → (∃𝑤𝑇 𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
132, 12ralimdaa 2987 . . . . . 6 (∀𝑧𝑇 𝒫 𝑧𝑇 → (∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤 → ∀𝑧𝑇 𝒫 𝑧𝑇))
1413imdistani 726 . . . . 5 ((∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤) → (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
15 r19.26 3093 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤))
16 r19.26 3093 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
1714, 15, 163imtr4i 281 . . . 4 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) → ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇))
18 ssid 3657 . . . . . . 7 𝒫 𝑧 ⊆ 𝒫 𝑧
19 sseq2 3660 . . . . . . . 8 (𝑤 = 𝒫 𝑧 → (𝒫 𝑧𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
2019rspcev 3340 . . . . . . 7 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤𝑇 𝒫 𝑧𝑤)
2118, 20mpan2 707 . . . . . 6 (𝒫 𝑧𝑇 → ∃𝑤𝑇 𝒫 𝑧𝑤)
2221anim2i 592 . . . . 5 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) → (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤))
2322ralimi 2981 . . . 4 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) → ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤))
2417, 23impbii 199 . . 3 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ↔ ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇))
2524anbi1i 731 . 2 ((∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)) ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)))
261, 25syl6bb 276 1 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  wcel 2030  wral 2941  wrex 2942  wss 3607  𝒫 cpw 4191   class class class wbr 4685  cen 7994  Tarskictsk 9608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-tsk 9609
This theorem is referenced by:  tskpw  9613  0tsk  9615  inttsk  9634  inatsk  9638
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