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Theorem tskpwss 9534
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 9532 . . . . 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 475 . . 3 (𝑇 ∈ Tarski → ∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦))
4 simpl 473 . . . 4 ((𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) → 𝒫 𝑥𝑇)
54ralimi 2948 . . 3 (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) → ∀𝑥𝑇 𝒫 𝑥𝑇)
63, 5syl 17 . 2 (𝑇 ∈ Tarski → ∀𝑥𝑇 𝒫 𝑥𝑇)
7 pweq 4139 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
87sseq1d 3617 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥𝑇 ↔ 𝒫 𝐴𝑇))
98rspccva 3298 . 2 ((∀𝑥𝑇 𝒫 𝑥𝑇𝐴𝑇) → 𝒫 𝐴𝑇)
106, 9sylan 488 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  wss 3560  𝒫 cpw 4136   class class class wbr 4623  cen 7912  Tarskictsk 9530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-tsk 9531
This theorem is referenced by:  tsksdom  9538  tskss  9540  tsktrss  9543  inttsk  9556  tskcard  9563
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