Theorem axunprim

Description: ax-un without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010)

	Ref	Expression
Assertion	axunprim	⊢ ¬ ∀ 𝑥 ¬ ∀ 𝑦 ( ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		axunnd	⊢ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )
2		df-an	⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ↔ ¬ ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) )
3	2	exbii	⊢ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ↔ ∃ 𝑥 ¬ ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) )
4		exnal	⊢ ( ∃ 𝑥 ¬ ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) ↔ ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) )
5	3 4	bitri	⊢ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ↔ ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) )
6	5	imbi1i	⊢ ( ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ( ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
7	6	albii	⊢ ( ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ( ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
8	7	exbii	⊢ ( ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ∃ 𝑥 ∀ 𝑦 ( ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
9		df-ex	⊢ ( ∃ 𝑥 ∀ 𝑦 ( ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ¬ ∀ 𝑥 ¬ ∀ 𝑦 ( ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
10	8 9	bitri	⊢ ( ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ¬ ∀ 𝑥 ¬ ∀ 𝑦 ( ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
11	1 10	mpbi	⊢ ¬ ∀ 𝑥 ¬ ∀ 𝑦 ( ¬ ∀ 𝑥 ( 𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )