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Metamath Proof Explorer

Theorem rabsssn

Description: Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019)

		Ref	Expression
	Assertion	rabsssn	⊢ ( { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ { 𝑋 } ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜑 → 𝑥 = 𝑋 ) )

Proof

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