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Metamath Proof Explorer

Theorem ditgex

Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014)

Ref Expression
Assertion ditgex ⊢ ∫ A B C dx ∈ V

Proof

Step Hyp Ref Expression
1 df-ditg ⊢ ∫ A B C dx = if A ≤ B ∫ A B C dx − ∫ B A C dx
2 itgex ⊢ ∫ A B C dx ∈ V
3 negex ⊢ − ∫ B A C dx ∈ V
4 2 3 ifex ⊢ if A ≤ B ∫ A B C dx − ∫ B A C dx ∈ V
5 1 4 eqeltri ⊢ ∫ A B C dx ∈ V