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Metamath Proof Explorer

Theorem rabsspr

Description: Conditions for a restricted class abstraction to be a subset of an unordered pair. (Contributed by Thierry Arnoux, 6-Jul-2025)

Ref Expression
Assertion rabsspr ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ { 𝑋 , 𝑌 } ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 df-rab ⊢ { 𝑥 ∈ 𝑉 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) }
2 dfpr2 ⊢ { 𝑋 , 𝑌 } = { 𝑥 ∣ ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) }
3 1 2 sseq12i ⊢ ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ { 𝑋 , 𝑌 } ↔ { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) } )
4 ss2ab ⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) } ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) )
5 impexp ⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) ↔ ( 𝑥 ∈ 𝑉 → ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) ) )
6 5 albii ⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑉 → ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) ) )
7 df-ral ⊢ ( ∀ 𝑥 ∈ 𝑉 ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑉 → ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) ) )
8 6 7 bitr4i ⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) )
9 3 4 8 3bitri ⊢ ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ { 𝑋 , 𝑌 } ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ) ) )