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 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)

Proof of Theorem tskpw

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltsk2g 10165	. . . . 5 ⊢ (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))))
2	1	ibi 269	. . . 4 ⊢ (𝑇 ∈ Tarski → (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇)))
3	2	simpld 497	. . 3 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇))
4		simpr 487	. . . 4 ⊢ ((𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) → 𝒫 𝑥 ∈ 𝑇)
5	4	ralimi 3158	. . 3 ⊢ (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) → ∀𝑥 ∈ 𝑇 𝒫 𝑥 ∈ 𝑇)
6	3, 5	syl 17	. 2 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝑇 𝒫 𝑥 ∈ 𝑇)
7		pweq 4540	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
8	7	eleq1d 2895	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑇 ↔ 𝒫 𝐴 ∈ 𝑇))
9	8	rspccva 3620	. 2 ⊢ ((∀𝑥 ∈ 𝑇 𝒫 𝑥 ∈ 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
10	6, 9	sylan 582	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1531   ∈ wcel 2108  ∀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 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  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 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  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  39912