MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eltsk2g Structured version   Visualization version   GIF version

Theorem eltsk2g 10790
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 10789 . 2 (𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
2 nfra1 3271 . . . . . . 7 𝑧𝑧𝑇 𝒫 𝑧𝑇
3 pweq 4620 . . . . . . . . . . . 12 (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
43sseq1d 4010 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝒫 𝑧𝑇 ↔ 𝒫 𝑤𝑇))
54rspccva 3606 . . . . . . . . . 10 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑤𝑇) → 𝒫 𝑤𝑇)
65adantlr 713 . . . . . . . . 9 (((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) ∧ 𝑤𝑇) → 𝒫 𝑤𝑇)
7 vpwex 5380 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
87elpw 4610 . . . . . . . . . 10 (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧𝑤)
9 ssel 3972 . . . . . . . . . 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 3144 . . . . . . 7 ((∀𝑧𝑇 𝒫 𝑧𝑇𝑧𝑇) → (∃𝑤𝑇 𝒫 𝑧𝑤 → 𝒫 𝑧𝑇))
132, 12ralimdaa 3247 . . . . . 6 (∀𝑧𝑇 𝒫 𝑧𝑇 → (∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤 → ∀𝑧𝑇 𝒫 𝑧𝑇))
1413imdistani 567 . . . . 5 ((∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤) → (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
15 r19.26 3100 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇𝑤𝑇 𝒫 𝑧𝑤))
16 r19.26 3100 . . . . 5 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ↔ (∀𝑧𝑇 𝒫 𝑧𝑇 ∧ ∀𝑧𝑇 𝒫 𝑧𝑇))
1714, 15, 163imtr4i 291 . . . 4 (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) → ∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇))
18 ssid 4001 . . . . . . 7 𝒫 𝑧 ⊆ 𝒫 𝑧
19 sseq2 4005 . . . . . . . 8 (𝑤 = 𝒫 𝑧 → (𝒫 𝑧𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
2019rspcev 3607 . . . . . . 7 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤𝑇 𝒫 𝑧𝑤)
2118, 20mpan2 689 . . . . . 6 (𝒫 𝑧𝑇 → ∃𝑤𝑇 𝒫 𝑧𝑤)
2221anim2i 615 . . . . 5 ((𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) → (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤))
2322ralimi 3072 . . . 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 622 . 2 ((∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)) ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇)))
261, 25bitrdi 286 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 394  wo 845  wcel 2098  wral 3050  wrex 3059  wss 3946  𝒫 cpw 4606   class class class wbr 5152  cen 8970  Tarskictsk 10787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-pow 5368
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-tsk 10788
This theorem is referenced by:  tskpw  10792  0tsk  10794  inttsk  10813  inatsk  10817
  Copyright terms: Public domain W3C validator