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Metamath Proof Explorer

Theorem tskop

Description: If A and B are members of a Tarski class, their ordered pair is also an element of the class. JFM CLASSES2 th. 4. (Contributed by FL, 22-Feb-2011)

Ref Expression
Assertion tskop ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑇 )

Proof

Step Hyp Ref Expression
1 dfopg ⊢ ( ( 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇 ) → ⟨ 𝐴 , 𝐵 ⟩ = { { 𝐴 } , { 𝐴 , 𝐵 } } )
2 1 3adant1 ⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇 ) → ⟨ 𝐴 , 𝐵 ⟩ = { { 𝐴 } , { 𝐴 , 𝐵 } } )
3 simp1 ⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇 ) → 𝑇 ∈ Tarski )
4 tsksn ⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ) → { 𝐴 } ∈ 𝑇 )
5 4 3adant3 ⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇 ) → { 𝐴 } ∈ 𝑇 )
6 tskpr ⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇 ) → { 𝐴 , 𝐵 } ∈ 𝑇 )
7 tskpr ⊢ ( ( 𝑇 ∈ Tarski ∧ { 𝐴 } ∈ 𝑇 ∧ { 𝐴 , 𝐵 } ∈ 𝑇 ) → { { 𝐴 } , { 𝐴 , 𝐵 } } ∈ 𝑇 )
8 3 5 6 7 syl3anc ⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇 ) → { { 𝐴 } , { 𝐴 , 𝐵 } } ∈ 𝑇 )
9 2 8 eqeltrd ⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑇 )