Theorem tskpwss 10766

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 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)

Proof of Theorem tskpwss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltskg 10764	. . . . 5 ⊢ (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))))
2	1	ibi 267	. . . 4 ⊢ (𝑇 ∈ Tarski → (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇)))
3	2	simpld 494	. . 3 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦))
4		simpl 482	. . . 4 ⊢ ((𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) → 𝒫 𝑥 ⊆ 𝑇)
5	4	ralimi 3073	. . 3 ⊢ (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) → ∀𝑥 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑇)
6	3, 5	syl 17	. 2 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑇)
7		pweq 4589	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
8	7	sseq1d 3990	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ⊆ 𝑇 ↔ 𝒫 𝐴 ⊆ 𝑇))
9	8	rspccva 3600	. 2 ⊢ ((∀𝑥 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)
10	6, 9	sylan 580	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  𝒫 cpw 4575   class class class wbr 5119   ≈ cen 8956  Tarskictsk 10762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-tsk 10763

This theorem is referenced by:  tsksdom  10770  tskss  10772  tsktrss  10775  inttsk  10788  tskcard  10795