Theorem iotassuni

Description: The iota class is a subset of the union of all elements satisfying ph . (Contributed by Mario Carneiro, 24-Dec-2016)

	Ref	Expression
Assertion	iotassuni	⊢ ( ℩ 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 }