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Theorem eltsk2g 9828
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 9827 . 2 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
2 nfra1 3088 . . . . . . 7 𝑧𝑧𝑇 𝒫 𝑧𝑇
3 pweq 4320 . . . . . . . . . . . 12 (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
43sseq1d 3794 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝒫 𝑧𝑇 ↔ 𝒫 𝑤𝑇))
54rspccva 3461 . . . . . . . . . 10 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑤𝑇) → 𝒫 𝑤𝑇)
65adantlr 706 . . . . . . . . 9 (((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) ∧ 𝑤𝑇) → 𝒫 𝑤𝑇)
7 vpwex 5015 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
87elpw 4323 . . . . . . . . . 10 (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧𝑤)
9 ssel 3757 . . . . . . . . . 10 (𝒫 𝑤𝑇 → (𝒫 𝑧 ∈ 𝒫 𝑤 → 𝒫 𝑧𝑇))
108, 9syl5bir 234 . . . . . . . . 9 (𝒫 𝑤𝑇 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
116, 10syl 17 . . . . . . . 8 (((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) ∧ 𝑤𝑇) → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
1211rexlimdva 3178 . . . . . . 7 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) → (∃𝑤𝑇 𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
132, 12ralimdaa 3105 . . . . . 6 (∀𝑧𝑇 𝒫 𝑧𝑇 → (∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤 → ∀𝑧𝑇 𝒫 𝑧𝑇))
1413imdistani 564 . . . . 5 ((∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤) → (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
15 r19.26 3211 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤))
16 r19.26 3211 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
1714, 15, 163imtr4i 283 . . . 4 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) → ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇))
18 ssid 3785 . . . . . . 7 𝒫 𝑧 ⊆ 𝒫 𝑧
19 sseq2 3789 . . . . . . . 8 (𝑤 = 𝒫 𝑧 → (𝒫 𝑧𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
2019rspcev 3462 . . . . . . 7 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤𝑇 𝒫 𝑧𝑤)
2118, 20mpan2 682 . . . . . 6 (𝒫 𝑧𝑇 → ∃𝑤𝑇 𝒫 𝑧𝑤)
2221anim2i 610 . . . . 5 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) → (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤))
2322ralimi 3099 . . . 4 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) → ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤))
2417, 23impbii 200 . . 3 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ↔ ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇))
2524anbi1i 617 . 2 ((∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)) ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)))
261, 25syl6bb 278 1 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873  wcel 2155  wral 3055  wrex 3056  wss 3734  𝒫 cpw 4317   class class class wbr 4811  cen 8159  Tarskictsk 9825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-pow 5003
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-tsk 9826
This theorem is referenced by:  tskpw  9830  0tsk  9832  inttsk  9851  inatsk  9855
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