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Theorem tskpwss 9780
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 9778 . . . . 5 (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))))
21ibi 256 . . . 4 (𝑇 ∈ Tarski → (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇)))
32simpld 482 . . 3 (𝑇 ∈ Tarski → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦))
4 simpl 468 . . . 4 ((𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) → 𝒫 𝑥𝑇)
54ralimi 3101 . . 3 (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) → ∀𝑥𝑇 𝒫 𝑥𝑇)
63, 5syl 17 . 2 (𝑇 ∈ Tarski → ∀𝑥𝑇 𝒫 𝑥𝑇)
7 pweq 4301 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
87sseq1d 3781 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥𝑇 ↔ 𝒫 𝐴𝑇))
98rspccva 3459 . 2 ((∀𝑥𝑇 𝒫 𝑥𝑇𝐴𝑇) → 𝒫 𝐴𝑇)
106, 9sylan 569 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 836   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723  𝒫 cpw 4298   class class class wbr 4787  cen 8110  Tarskictsk 9776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-tsk 9777
This theorem is referenced by:  tsksdom  9784  tskss  9786  tsktrss  9789  inttsk  9802  tskcard  9809
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