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

Proof

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