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	$$⊢\left(\iota x|\phi \right)\subseteq \bigcup \left\{x|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		iotauni2	$$⊢\exists y\left\{x|\phi \right\}=\left\{y\right\}\to \left(\iota x|\phi \right)=\bigcup \left\{x|\phi \right\}$$
2		eqimss	$$⊢\left(\iota x|\phi \right)=\bigcup \left\{x|\phi \right\}\to \left(\iota x|\phi \right)\subseteq \bigcup \left\{x|\phi \right\}$$
3	1 2	syl	$$⊢\exists y\left\{x|\phi \right\}=\left\{y\right\}\to \left(\iota x|\phi \right)\subseteq \bigcup \left\{x|\phi \right\}$$
4		iotanul2	$$⊢\neg \exists y\left\{x|\phi \right\}=\left\{y\right\}\to \left(\iota x|\phi \right)=\varnothing$$
5		0ss	$$⊢\varnothing \subseteq \bigcup \left\{x|\phi \right\}$$
6	4 5	eqsstrdi	$$⊢\neg \exists y\left\{x|\phi \right\}=\left\{y\right\}\to \left(\iota x|\phi \right)\subseteq \bigcup \left\{x|\phi \right\}$$
7	3 6	pm2.61i	$$⊢\left(\iota x|\phi \right)\subseteq \bigcup \left\{x|\phi \right\}$$