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Theorem eltsk2g 9517
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 9516 . 2 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
2 nfra1 2936 . . . . . . 7 𝑧𝑧𝑇 𝒫 𝑧𝑇
3 pweq 4133 . . . . . . . . . . . 12 (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
43sseq1d 3611 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝒫 𝑧𝑇 ↔ 𝒫 𝑤𝑇))
54rspccva 3294 . . . . . . . . . 10 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑤𝑇) → 𝒫 𝑤𝑇)
65adantlr 750 . . . . . . . . 9 (((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) ∧ 𝑤𝑇) → 𝒫 𝑤𝑇)
7 vpwex 4809 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
87elpw 4136 . . . . . . . . . 10 (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧𝑤)
9 ssel 3577 . . . . . . . . . 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 3024 . . . . . . 7 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) → (∃𝑤𝑇 𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
132, 12ralimdaa 2952 . . . . . 6 (∀𝑧𝑇 𝒫 𝑧𝑇 → (∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤 → ∀𝑧𝑇 𝒫 𝑧𝑇))
1413imdistani 725 . . . . 5 ((∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤) → (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
15 r19.26 3057 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤))
16 r19.26 3057 . . . . 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 3603 . . . . . . 7 𝒫 𝑧 ⊆ 𝒫 𝑧
19 sseq2 3606 . . . . . . . 8 (𝑤 = 𝒫 𝑧 → (𝒫 𝑧𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
2019rspcev 3295 . . . . . . 7 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤𝑇 𝒫 𝑧𝑤)
2118, 20mpan2 706 . . . . . 6 (𝒫 𝑧𝑇 → ∃𝑤𝑇 𝒫 𝑧𝑤)
2221anim2i 592 . . . . 5 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) → (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤))
2322ralimi 2947 . . . 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 730 . 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 383  wa 384  wcel 1987  wral 2907  wrex 2908  wss 3555  𝒫 cpw 4130   class class class wbr 4613  cen 7896  Tarskictsk 9514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-pow 4803
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-tsk 9515
This theorem is referenced by:  tskpw  9519  0tsk  9521  inttsk  9540  inatsk  9544
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