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Metamath Proof Explorer

Theorem ssrabeq

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

Ref Expression
Assertion ssrabeq ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } )

Proof

Step Hyp Ref Expression
1 ssrab2 ⊢ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ 𝑉
2 1 biantru ⊢ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ∧ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ 𝑉 ) )
3 eqss ⊢ ( 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ∧ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ 𝑉 ) )
4 2 3 bitr4i ⊢ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } )