Theorem elALTtco 36669

Description: Derivation of el 5383 from ax-tco 36660. Use el 5383 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elALTtco	⊢ ∃𝑦 𝑥 ∈ 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem elALTtco

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-tco 36660	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
2		simpl 482	. 2 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))) → 𝑥 ∈ 𝑦)
3	1, 2	eximii 1839	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-tco 36660

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by: (None)