Theorem tskpw 10822

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 10820	. . . . 5 ⊢ (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))))
2	1	ibi 267	. . . 4 ⊢ (𝑇 ∈ Tarski → (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇)))
3	2	simpld 494	. . 3 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇))
4		simpr 484	. . . 4 ⊢ ((𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) → 𝒫 𝑥 ∈ 𝑇)
5	4	ralimi 3089	. . 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 4636	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
8	7	eleq1d 2829	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑇 ↔ 𝒫 𝐴 ∈ 𝑇))
9	8	rspccva 3634	. 2 ⊢ ((∀𝑥 ∈ 𝑇 𝒫 𝑥 ∈ 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
10	6, 9	sylan 579	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166   ≈ cen 9000  Tarskictsk 10817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-pow 5383

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-tsk 10818

This theorem is referenced by:  tsksn  10829  tsksuc  10831  tskr1om  10836  inttsk  10843  tskcard  10850  tskwun  10853  grutsk1  10890  pwinfi3  43525