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Theorem tskpw 9828
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 9826 . . . . 5 (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))))
21ibi 258 . . . 4 (𝑇 ∈ Tarski → (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇)))
32simpld 488 . . 3 (𝑇 ∈ Tarski → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇))
4 simpr 477 . . . 4 ((𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇) → 𝒫 𝑥𝑇)
54ralimi 3099 . . 3 (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ 𝒫 𝑥𝑇) → ∀𝑥𝑇 𝒫 𝑥𝑇)
63, 5syl 17 . 2 (𝑇 ∈ Tarski → ∀𝑥𝑇 𝒫 𝑥𝑇)
7 pweq 4318 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
87eleq1d 2829 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥𝑇 ↔ 𝒫 𝐴𝑇))
98rspccva 3460 . 2 ((∀𝑥𝑇 𝒫 𝑥𝑇𝐴𝑇) → 𝒫 𝐴𝑇)
106, 9sylan 575 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 873   = wceq 1652  wcel 2155  wral 3055  wss 3732  𝒫 cpw 4315   class class class wbr 4809  cen 8157  Tarskictsk 9823
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 4941  ax-pow 5001
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 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-tsk 9824
This theorem is referenced by:  tsksn  9835  tsksuc  9837  tskr1om  9842  inttsk  9849  tskcard  9856  tskwun  9859  grutsk1  9896  pwinfi3  38543
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