Theorem exel 5373

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

This theorem looks similar to el 5377, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1816, ax-6 1974, and ax-pr 5362. 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 5374. (Contributed by SN, 23-Dec-2024.)

Assertion

Ref	Expression
exel	⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem exel

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pr 5362	. 2 ⊢ ∃𝑦∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦)
2		ax6ev 1976	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑧
3		pm2.07 908	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑧 ∨ 𝑥 = 𝑧))
4	2, 3	eximii 1844	. . 3 ⊢ ∃𝑥(𝑥 = 𝑧 ∨ 𝑥 = 𝑧)
5		exim 1841	. . 3 ⊢ (∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦) → (∃𝑥(𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → ∃𝑥 𝑥 ∈ 𝑦))
6	4, 5	mpi 20	. 2 ⊢ (∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦) → ∃𝑥 𝑥 ∈ 𝑦)
7	1, 6	eximii 1844	1 ⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦

Syntax hints:   → wi 4   ∨ wo 853  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-6 1974  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-or 854  df-ex 1787

This theorem is referenced by:  exexneq  5374