Theorem iotassuni

Description: The iota class is a subset of the union of all elements satisfying ph . (Contributed by Mario Carneiro, 24-Dec-2016) Remove dependency on ax-10 , ax-11 , ax-12 . (Revised by SN, 6-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		iotauni2	⊢ ( ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) = ∪ { 𝑥 ∣ 𝜑 } )
2		eqimss	⊢ ( ( ℩ 𝑥 𝜑 ) = ∪ { 𝑥 ∣ 𝜑 } → ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 } )
3	1 2	syl	⊢ ( ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 } )
4		iotanul2	⊢ ( ¬ ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) = ∅ )
5		0ss	⊢ ∅ ⊆ ∪ { 𝑥 ∣ 𝜑 }
6	4 5	eqsstrdi	⊢ ( ¬ ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 } )
7	3 6	pm2.61i	⊢ ( ℩ 𝑥 𝜑 ) ⊆ ∪ { 𝑥 ∣ 𝜑 }