Theorem el

Description: Every set is an element of some other set. See elALT for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	el	⊢ ∃ 𝑦 𝑥 ∈ 𝑦

Proof

Step	Hyp	Ref	Expression
1		zfpow	⊢ ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 )
2		ax9	⊢ ( 𝑧 = 𝑥 → ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) )
3	2	alrimiv	⊢ ( 𝑧 = 𝑥 → ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) )
4		ax8	⊢ ( 𝑧 = 𝑥 → ( 𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦 ) )
5	3 4	embantd	⊢ ( 𝑧 = 𝑥 → ( ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) → 𝑥 ∈ 𝑦 ) )
6	5	spimvw	⊢ ( ∀ 𝑧 ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) → 𝑥 ∈ 𝑦 )
7	1 6	eximii	⊢ ∃ 𝑦 𝑥 ∈ 𝑦