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Theorem tskpwss 10746
Description: First axiom of a Tarski class. The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskpwss ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)

Proof of Theorem tskpwss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eltskg 10744 . . . . 5 (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))))
21ibi 267 . . . 4 (𝑇 ∈ Tarski → (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇)))
32simpld 494 . . 3 (𝑇 ∈ Tarski → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦))
4 simpl 482 . . . 4 ((𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) → 𝒫 𝑥𝑇)
54ralimi 3077 . . 3 (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) → ∀𝑥𝑇 𝒫 𝑥𝑇)
63, 5syl 17 . 2 (𝑇 ∈ Tarski → ∀𝑥𝑇 𝒫 𝑥𝑇)
7 pweq 4611 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
87sseq1d 4008 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥𝑇 ↔ 𝒫 𝐴𝑇))
98rspccva 3605 . 2 ((∀𝑥𝑇 𝒫 𝑥𝑇𝐴𝑇) → 𝒫 𝐴𝑇)
106, 9sylan 579 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 844   = wceq 1533  wcel 2098  wral 3055  wrex 3064  wss 3943  𝒫 cpw 4597   class class class wbr 5141  cen 8935  Tarskictsk 10742
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-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-tsk 10743
This theorem is referenced by:  tsksdom  10750  tskss  10752  tsktrss  10755  inttsk  10768  tskcard  10775
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