Theorem exel 5453

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

This theorem looks similar to el 5457, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1807, ax-6 1967, and ax-pr 5447. 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 5454. (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 5447	. 2 ⊢ ∃𝑦∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦)
2		ax6ev 1969	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑧
3		pm2.07 901	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑧 ∨ 𝑥 = 𝑧))
4	2, 3	eximii 1835	. . 3 ⊢ ∃𝑥(𝑥 = 𝑧 ∨ 𝑥 = 𝑧)
5		exim 1832	. . 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 1835	1 ⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦

Syntax hints:   → wi 4   ∨ wo 846  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-6 1967  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-or 847  df-ex 1778

This theorem is referenced by:  exexneq  5454