Theorem exel 5444

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

This theorem looks similar to el 5448, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1806, ax-6 1965, and ax-pr 5438. 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 5445. (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 5438	. 2 ⊢ ∃𝑦∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦)
2		ax6ev 1967	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑧
3		pm2.07 902	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑧 ∨ 𝑥 = 𝑧))
4	2, 3	eximii 1834	. . 3 ⊢ ∃𝑥(𝑥 = 𝑧 ∨ 𝑥 = 𝑧)
5		exim 1831	. . 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 1834	1 ⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦

Syntax hints:   → wi 4   ∨ wo 847  ∀wal 1535  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-6 1965  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1777

This theorem is referenced by:  exexneq  5445