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Metamath Proof Explorer

Theorem tpid2

Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994) (Proof shortened by Andrew Salmon, 29-Jun-2011)

Ref Expression
Hypothesis tpid2.1 ⊢ B ∈ V
Assertion tpid2 ⊢ B ∈ A B C

Proof

Step Hyp Ref Expression
1 tpid2.1 ⊢ B ∈ V
2 eqid ⊢ B = B
3 2 3mix2i ⊢ B = A ∨ B = B ∨ B = C
4 1 eltp ⊢ B ∈ A B C ↔ B = A ∨ B = B ∨ B = C
5 3 4 mpbir ⊢ B ∈ A B C