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)
Ref | Expression | ||
---|---|---|---|
Assertion | tsktrss | ⊢ ( ( 𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇 ) → 𝐴 ⊆ 𝑇 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 | ⊢ ( ( 𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇 ) → Tr 𝐴 ) | |
2 | dftr4 | ⊢ ( Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴 ) | |
3 | 1 2 | sylib | ⊢ ( ( 𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇 ) → 𝐴 ⊆ 𝒫 𝐴 ) |
4 | tskpwss | ⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ) → 𝒫 𝐴 ⊆ 𝑇 ) | |
5 | 4 | 3adant2 | ⊢ ( ( 𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇 ) → 𝒫 𝐴 ⊆ 𝑇 ) |
6 | 3 5 | sstrd | ⊢ ( ( 𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇 ) → 𝐴 ⊆ 𝑇 ) |