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Metamath Proof Explorer

Theorem trin

Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994)

Ref Expression
Assertion trin ( ( Tr 𝐴 ∧ Tr 𝐵 ) → Tr ( 𝐴 ∩ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 elin ⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
2 trss ⊢ ( Tr 𝐴 → ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴 ) )
3 trss ⊢ ( Tr 𝐵 → ( 𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵 ) )
4 2 3 im2anan9 ⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ) )
5 1 4 biimtrid ⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) → ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ) )
6 ssin ⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ↔ 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) )
7 5 6 imbitrdi ⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) → 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) ) )
8 7 ralrimiv ⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) )
9 dftr3 ⊢ ( Tr ( 𝐴 ∩ 𝐵 ) ↔ ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) )
10 8 9 sylibr ⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → Tr ( 𝐴 ∩ 𝐵 ) )