Theorem exel

Description: There exist two sets, one a member of the other.

This theorem looks similar to el , but its meaning is different. It only depends on the axioms ax-mp to ax-4 , ax-6 , and ax-pr . This theorem does not exclude that these two sets could actually be one single set containing itself. That two different sets exist is proved by exexneq . (Contributed by SN, 23-Dec-2024)

	Ref	Expression
Assertion	exel	⊢ ∃ 𝑦 ∃ 𝑥 𝑥 ∈ 𝑦

Proof

Step	Hyp	Ref	Expression
1		ax-pr	⊢ ∃ 𝑦 ∀ 𝑥 ( ( 𝑥 = 𝑧 ∨ 𝑥 = 𝑧 ) → 𝑥 ∈ 𝑦 )
2		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑧
3		pm2.07	⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝑧 ∨ 𝑥 = 𝑧 ) )
4	2 3	eximii	⊢ ∃ 𝑥 ( 𝑥 = 𝑧 ∨ 𝑥 = 𝑧 )
5		exim	⊢ ( ∀ 𝑥 ( ( 𝑥 = 𝑧 ∨ 𝑥 = 𝑧 ) → 𝑥 ∈ 𝑦 ) → ( ∃ 𝑥 ( 𝑥 = 𝑧 ∨ 𝑥 = 𝑧 ) → ∃ 𝑥 𝑥 ∈ 𝑦 ) )
6	4 5	mpi	⊢ ( ∀ 𝑥 ( ( 𝑥 = 𝑧 ∨ 𝑥 = 𝑧 ) → 𝑥 ∈ 𝑦 ) → ∃ 𝑥 𝑥 ∈ 𝑦 )
7	1 6	eximii	⊢ ∃ 𝑦 ∃ 𝑥 𝑥 ∈ 𝑦