Theorem sn-iotassuni

Description: iotassuni without ax-10 , ax-11 , ax-12 . (Contributed by SN, 6-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		sn-iotauni	⊢ ( ∃ 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		sn-iotanul	⊢ ( ¬ ∃ 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 }