Theorem axinfprim

Description: ax-inf without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010)

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

Proof

Step	Hyp	Ref	Expression
1		axinfnd	⊢ ∃ 𝑥 ( 𝑦 ∈ 𝑧 → ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ) )
2		df-an	⊢ ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ↔ ¬ ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) )
3	2	exbii	⊢ ( ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ↔ ∃ 𝑧 ¬ ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) )
4		exnal	⊢ ( ∃ 𝑧 ¬ ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ↔ ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) )
5	3 4	bitri	⊢ ( ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ↔ ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) )
6	5	imbi2i	⊢ ( ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ↔ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) )
7	6	albii	⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) )
8	7	anbi2i	⊢ ( ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ) ↔ ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) )
9		df-an	⊢ ( ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) ↔ ¬ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) )
10	8 9	bitri	⊢ ( ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ) ↔ ¬ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) )
11	10	imbi2i	⊢ ( ( 𝑦 ∈ 𝑧 → ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ) ) ↔ ( 𝑦 ∈ 𝑧 → ¬ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) ) )
12	11	exbii	⊢ ( ∃ 𝑥 ( 𝑦 ∈ 𝑧 → ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ) ) ↔ ∃ 𝑥 ( 𝑦 ∈ 𝑧 → ¬ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) ) )
13		df-ex	⊢ ( ∃ 𝑥 ( 𝑦 ∈ 𝑧 → ¬ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) ) ↔ ¬ ∀ 𝑥 ¬ ( 𝑦 ∈ 𝑧 → ¬ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) ) )
14	12 13	bitri	⊢ ( ∃ 𝑥 ( 𝑦 ∈ 𝑧 → ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ) ) ↔ ¬ ∀ 𝑥 ¬ ( 𝑦 ∈ 𝑧 → ¬ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) ) )
15	1 14	mpbi	⊢ ¬ ∀ 𝑥 ¬ ( 𝑦 ∈ 𝑧 → ¬ ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ ∀ 𝑧 ( 𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥 ) ) ) )