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Metamath Proof Explorer

Theorem el

Description: Any set is an element of some other set. See elALT for a shorter proof using more axioms, and see elALT2 for a proof that uses ax-9 and ax-pow instead of ax-pr . (Contributed by NM, 4-Jan-2002) (Proof shortened by Andrew Salmon, 25-Jul-2011) Use ax-pr instead of ax-9 and ax-pow . (Revised by BTernaryTau, 2-Dec-2024) (Proof shortened by Matthew House, 6-Apr-2026)

		Ref	Expression
	Assertion	el	$⊢ \exists y x \in y$

Proof

Step	Hyp	Ref	Expression
1		ax-pr	$⊢ \exists y \forall z ((z = x \lor z = x) \to z \in y)$
2		orc	$⊢ z = x \to (z = x \lor z = x)$
3		ax8v1	$⊢ z = x \to (z \in y \to x \in y)$
4	2 3	embantd	$⊢ z = x \to (((z = x \lor z = x) \to z \in y) \to x \in y)$
5	4	spimvw	$⊢ \forall z ((z = x \lor z = x) \to z \in y) \to x \in y$
6	1 5	eximii	$⊢ \exists y x \in y$