Theorem elALTtco 37020

Description: Derivation of el 5418 from ax-tco 37011. Use el 5418 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 37011	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
2		simpl 487	. 2 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))) → 𝑥 ∈ 𝑦)
3	1, 2	eximii 1866	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-tco 37011

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809

This theorem is used by: (None)