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

Theorem tskpw 10167
Description: Second axiom of a Tarski class. The powerset of an element of a Tarski class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskpw ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)

Proof of Theorem tskpw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eltsk2g 10165 . . . . 5 (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))))
21ibi 269 . . . 4 (𝑇 ∈ Tarski → (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇)))
32simpld 497 . . 3 (𝑇 ∈ Tarski → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇))
4 simpr 487 . . . 4 ((𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇) → 𝒫 𝑥𝑇)
54ralimi 3158 . . 3 (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇) → ∀𝑥𝑇 𝒫 𝑥𝑇)
63, 5syl 17 . 2 (𝑇 ∈ Tarski → ∀𝑥𝑇 𝒫 𝑥𝑇)
7 pweq 4540 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
87eleq1d 2895 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥𝑇 ↔ 𝒫 𝐴𝑇))
98rspccva 3620 . 2 ((∀𝑥𝑇 𝒫 𝑥𝑇𝐴𝑇) → 𝒫 𝐴𝑇)
106, 9sylan 582 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843   = wceq 1530  wcel 2107  wral 3136  wss 3934  𝒫 cpw 4537   class class class wbr 5057  cen 8498  Tarskictsk 10162
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-pow 5257
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-tsk 10163
This theorem is referenced by:  tsksn  10174  tsksuc  10176  tskr1om  10181  inttsk  10188  tskcard  10195  tskwun  10198  grutsk1  10235  pwinfi3  39907
  Copyright terms: Public domain W3C validator