Theorem tskss 10760

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 5306	. . . 4 ⊢ (𝐴 ∈ 𝑇 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
2	1	adantl 487	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
3		tskpwss 10754	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)
4	3	sseld 3937	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐵 ∈ 𝒫 𝐴 → 𝐵 ∈ 𝑇))
5	2, 4	sylbird 263	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐵 ⊆ 𝐴 → 𝐵 ∈ 𝑇))
6	5	3impia 1135	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑇)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146   ⊆ wss 3906  𝒫 cpw 4564  Tarskictsk 10750

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-tsk 10751

This theorem is used by:  tskin  10761  tsksn  10762  tsksuc  10764  tsk0  10765  tskr1om2  10770  tskint  10787