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Metamath Proof Explorer

Theorem difrab0eq

Description: If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017)

Ref Expression
Assertion difrab0eq ⊢ V ∖ x ∈ V | φ = ∅ ↔ V = x ∈ V | φ

Proof

Step Hyp Ref Expression
1 ssdif0 ⊢ V ⊆ x ∈ V | φ ↔ V ∖ x ∈ V | φ = ∅
2 ssrabeq ⊢ V ⊆ x ∈ V | φ ↔ V = x ∈ V | φ
3 1 2 bitr3i ⊢ V ∖ x ∈ V | φ = ∅ ↔ V = x ∈ V | φ