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Theorem eltsk2g 10695
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 10694 . 2 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
2 nfra1 3266 . . . . . . 7 𝑧𝑧𝑇 𝒫 𝑧𝑇
3 pweq 4578 . . . . . . . . . . . 12 (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
43sseq1d 3979 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝒫 𝑧𝑇 ↔ 𝒫 𝑤𝑇))
54rspccva 3582 . . . . . . . . . 10 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑤𝑇) → 𝒫 𝑤𝑇)
65adantlr 714 . . . . . . . . 9 (((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) ∧ 𝑤𝑇) → 𝒫 𝑤𝑇)
7 vpwex 5336 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
87elpw 4568 . . . . . . . . . 10 (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧𝑤)
9 ssel 3941 . . . . . . . . . 10 (𝒫 𝑤𝑇 → (𝒫 𝑧 ∈ 𝒫 𝑤 → 𝒫 𝑧𝑇))
108, 9biimtrrid 242 . . . . . . . . 9 (𝒫 𝑤𝑇 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
116, 10syl 17 . . . . . . . 8 (((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) ∧ 𝑤𝑇) → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
1211rexlimdva 3149 . . . . . . 7 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) → (∃𝑤𝑇 𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
132, 12ralimdaa 3242 . . . . . 6 (∀𝑧𝑇 𝒫 𝑧𝑇 → (∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤 → ∀𝑧𝑇 𝒫 𝑧𝑇))
1413imdistani 570 . . . . 5 ((∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤) → (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
15 r19.26 3111 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤))
16 r19.26 3111 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
1714, 15, 163imtr4i 292 . . . 4 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) → ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇))
18 ssid 3970 . . . . . . 7 𝒫 𝑧 ⊆ 𝒫 𝑧
19 sseq2 3974 . . . . . . . 8 (𝑤 = 𝒫 𝑧 → (𝒫 𝑧𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
2019rspcev 3583 . . . . . . 7 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤𝑇 𝒫 𝑧𝑤)
2118, 20mpan2 690 . . . . . 6 (𝒫 𝑧𝑇 → ∃𝑤𝑇 𝒫 𝑧𝑤)
2221anim2i 618 . . . . 5 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) → (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤))
2322ralimi 3083 . . . 4 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) → ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤))
2417, 23impbii 208 . . 3 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ↔ ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇))
2524anbi1i 625 . 2 ((∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)) ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)))
261, 25bitrdi 287 1 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  wcel 2107  wral 3061  wrex 3070  wss 3914  𝒫 cpw 4564   class class class wbr 5109  cen 8886  Tarskictsk 10692
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-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-pow 5324
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-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-tsk 10693
This theorem is referenced by:  tskpw  10697  0tsk  10699  inttsk  10718  inatsk  10722
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