Theorem exel 5434

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

This theorem looks similar to el 5438, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1812, ax-6 1972, and ax-pr 5428. 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 5435. (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 5428	. 2 ⊢ ∃𝑦∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦)
2		ax6ev 1974	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑧
3		pm2.07 902	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑧 ∨ 𝑥 = 𝑧))
4	2, 3	eximii 1840	. . 3 ⊢ ∃𝑥(𝑥 = 𝑧 ∨ 𝑥 = 𝑧)
5		exim 1837	. . 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 1840	1 ⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦

Syntax hints:   → wi 4   ∨ wo 846  ∀wal 1540  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-6 1972  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-or 847  df-ex 1783

This theorem is referenced by:  exexneq  5435