Theorem axnulALT3 35502

Description: Alternate proof of axnul 5268, proved from propositional calculus, ax-gen 1825, ax-4 1839, ax-5 1940, and ax-inf2 9606. (Contributed by BTernaryTau, 22-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axnulALT3	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem axnulALT3

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

Step	Hyp	Ref	Expression
1		exsimpr 1899	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦 ¬ 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
2		ax-inf2 9606	. . 3 ⊢ ∃𝑧(∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦 ¬ 𝑦 ∈ 𝑥) ∧ ∀𝑥(𝑥 ∈ 𝑧 → ∃𝑦(𝑦 ∈ 𝑧 ∧ ∀𝑤(𝑤 ∈ 𝑦 ↔ (𝑤 ∈ 𝑥 ∨ 𝑤 = 𝑥)))))
3		simpl 487	. . 3 ⊢ ((∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦 ¬ 𝑦 ∈ 𝑥) ∧ ∀𝑥(𝑥 ∈ 𝑧 → ∃𝑦(𝑦 ∈ 𝑧 ∧ ∀𝑤(𝑤 ∈ 𝑦 ↔ (𝑤 ∈ 𝑥 ∨ 𝑤 = 𝑥))))) → ∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦 ¬ 𝑦 ∈ 𝑥))
4	2, 3	eximii 1867	. 2 ⊢ ∃𝑧∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
5	1, 4	exlimiiv 1961	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by: (None)