Theorem tskpwss 9453

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 9451	. . . . 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 255	. . . 4 ⊢ (𝑇 ∈ Tarski → (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇)))
3	2	simpld 474	. . 3 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦))
4		simpl 472	. . . 4 ⊢ ((𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) → 𝒫 𝑥 ⊆ 𝑇)
5	4	ralimi 2936	. . 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 4111	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
8	7	sseq1d 3595	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ⊆ 𝑇 ↔ 𝒫 𝐴 ⊆ 𝑇))
9	8	rspccva 3281	. 2 ⊢ ((∀𝑥 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)
10	6, 9	sylan 487	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ≈ cen 7838  Tarskictsk 9449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-tsk 9450

This theorem is referenced by:  tsksdom  9457  tskss  9459  tsktrss  9462  inttsk  9475  tskcard  9482