Theorem tsktrss 10799

Description: A transitive element of a Tarski class is a part of the class. JFM CLASSES2 th. 8. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tsktrss	⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)

Proof of Theorem tsktrss

Step	Hyp	Ref	Expression
1		simp2 1136	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → Tr 𝐴)
2		dftr4 5272	. . 3 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
3	1, 2	sylib 218	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝒫 𝐴)
4		tskpwss 10790	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)
5	4	3adant2 1130	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)
6	3, 5	sstrd 4006	1 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2106   ⊆ wss 3963  𝒫 cpw 4605  Tr wtr 5265  Tarskictsk 10786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-tr 5266  df-tsk 10787

This theorem is referenced by: (None)