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	⊢ ∃ 𝑦 𝑥 ∈ 𝑦

Proof

Step	Hyp	Ref	Expression
1		ax-pr	⊢ ∃ 𝑦 ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) → 𝑧 ∈ 𝑦 )
2		orc	⊢ ( 𝑧 = 𝑥 → ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) )
3		ax8v1	⊢ ( 𝑧 = 𝑥 → ( 𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦 ) )
4	2 3	embantd	⊢ ( 𝑧 = 𝑥 → ( ( ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) → 𝑧 ∈ 𝑦 ) → 𝑥 ∈ 𝑦 ) )
5	4	spimvw	⊢ ( ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 = 𝑥 ) → 𝑧 ∈ 𝑦 ) → 𝑥 ∈ 𝑦 )
6	1 5	eximii	⊢ ∃ 𝑦 𝑥 ∈ 𝑦