Theorem rabsstp

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

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

Proof

Step	Hyp	Ref	Expression
1		df-rab	⊢ { 𝑥 ∈ 𝑉 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) }
2		dftp2	⊢ { 𝑋 , 𝑌 , 𝑍 } = { 𝑥 ∣ ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) }
3	1 2	sseq12i	⊢ ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ { 𝑋 , 𝑌 , 𝑍 } ↔ { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) } )
4		ss2ab	⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) } ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) )
5		impexp	⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) ↔ ( 𝑥 ∈ 𝑉 → ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) ) )
6	5	albii	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑉 → ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) ) )
7		df-ral	⊢ ( ∀ 𝑥 ∈ 𝑉 ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑉 → ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) ) )
8	6 7	bitr4i	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) )
9	3 4 8	3bitri	⊢ ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ { 𝑋 , 𝑌 , 𝑍 } ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜑 → ( 𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍 ) ) )