Theorem elALTtco 37191

Description: Derivation of el 5405 from ax-tco 37182. Use el 5405 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 37182	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
2		simpl 488	. 2 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))) → 𝑥 ∈ 𝑦)
3	1, 2	eximii 1870	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-tco 37182

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)