Theorem tskpw 10708

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 10706	. . . . 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 498	. . 3 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇))
4		simpr 488	. . . 4 ⊢ ((𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) → 𝒫 𝑥 ∈ 𝑇)
5	4	ralimi 3098	. . 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 4568	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
8	7	eleq1d 2846	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑇 ↔ 𝒫 𝐴 ∈ 𝑇))
9	8	rspccva 3580	. 2 ⊢ ((∀𝑥 ∈ 𝑇 𝒫 𝑥 ∈ 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
10	6, 9	sylan 589	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3904  𝒫 cpw 4554   class class class wbr 5099   ≈ cen 8920  Tarskictsk 10703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pow 5321

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-tsk 10704

This theorem is referenced by:  tsksn  10715  tsksuc  10717  tskr1om  10722  inttsk  10729  tskcard  10736  tskwun  10739  grutsk1  10776  pwinfi3  44103