Theorem iotassuniOLD

Description: Obsolete version of iotassuni as of 23-Dec-2024. (Contributed by Mario Carneiro, 24-Dec-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	iotassuniOLD	⊢ ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		iotauni	⊢ ( ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) = ∪ { 𝑥 ∣ 𝜑 } )
2		eqimss	⊢ ( ( ℩ 𝑥 𝜑 ) = ∪ { 𝑥 ∣ 𝜑 } → ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 } )
3	1 2	syl	⊢ ( ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 } )
4		iotanul	⊢ ( ¬ ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) = ∅ )
5		0ss	⊢ ∅ ⊆ ∪ { 𝑥 ∣ 𝜑 }
6	4 5	eqsstrdi	⊢ ( ¬ ∃! 𝑥 𝜑 → ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 } )
7	3 6	pm2.61i	⊢ ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 }