Theorem tskpw 9781

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 9779	. . . . 5 ⊢ (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))))
2	1	ibi 256	. . . 4 ⊢ (𝑇 ∈ Tarski → (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇)))
3	2	simpld 482	. . 3 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇))
4		simpr 471	. . . 4 ⊢ ((𝒫 𝑥 ⊆ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇) → 𝒫 𝑥 ∈ 𝑇)
5	4	ralimi 3101	. . 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 4301	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
8	7	eleq1d 2835	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑇 ↔ 𝒫 𝐴 ∈ 𝑇))
9	8	rspccva 3459	. 2 ⊢ ((∀𝑥 ∈ 𝑇 𝒫 𝑥 ∈ 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
10	6, 9	sylan 569	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 382   ∨ wo 836   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ⊆ 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  ax-sep 4916  ax-pow 4975

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:  tsksn  9788  tsksuc  9790  tskr1om  9795  inttsk  9802  tskcard  9809  tskwun  9812  grutsk1  9849  pwinfi3  38394