Theorem axpowprim

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

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

Proof

Step	Hyp	Ref	Expression
1		axpownd	⊢ ( ¬ 𝑥 = 𝑦 → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
2		df-ex	⊢ ( ∃ 𝑧 𝑥 ∈ 𝑦 ↔ ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 )
3	2	imbi1i	⊢ ( ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) ↔ ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) )
4	3	albii	⊢ ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) ↔ ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) )
5	4	imbi1i	⊢ ( ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ( ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
6	5	albii	⊢ ( ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ( ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
7	6	exbii	⊢ ( ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
8		df-ex	⊢ ( ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ¬ ∀ 𝑥 ¬ ∀ 𝑦 ( ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
9	7 8	bitri	⊢ ( ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ¬ ∀ 𝑥 ¬ ∀ 𝑦 ( ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
10	1 9	sylib	⊢ ( ¬ 𝑥 = 𝑦 → ¬ ∀ 𝑥 ¬ ∀ 𝑦 ( ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
11	10	con4i	⊢ ( ∀ 𝑥 ¬ ∀ 𝑦 ( ∀ 𝑥 ( ¬ ∀ 𝑧 ¬ 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) → 𝑥 = 𝑦 )