Theorem tskss 10757

Description: The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)

Assertion

Ref	Expression
tskss	⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑇)

Proof of Theorem tskss

Step	Hyp	Ref	Expression
1		elpw2g 5345	. . . 4 ⊢ (𝐴 ∈ 𝑇 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
2	1	adantl 480	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
3		tskpwss 10751	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)
4	3	sseld 3982	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐵 ∈ 𝒫 𝐴 → 𝐵 ∈ 𝑇))
5	2, 4	sylbird 259	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐵 ⊆ 𝐴 → 𝐵 ∈ 𝑇))
6	5	3impia 1115	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑇)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   ∈ wcel 2104   ⊆ wss 3949  𝒫 cpw 4603  Tarskictsk 10747

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-tsk 10748

This theorem is referenced by:  tskin  10758  tsksn  10759  tsksuc  10761  tsk0  10762  tskr1om2  10767  tskint  10784