Theorem dffr6

Description: Alternate definition of df-fr . See dffr5 for a definition without dummy variables (but note that their equivalence uses ax-sep ). (Contributed by BJ, 16-Nov-2024)

	Ref	Expression
Assertion	dffr6	⊢ ( 𝑅 Fr 𝐴 ↔ ∀ 𝑥 ∈ ( 𝒫 𝐴 ∖ { ∅ } ) ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		velpw	⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 )
2	1	bicomi	⊢ ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ 𝒫 𝐴 )
3		velsn	⊢ ( 𝑥 ∈ { ∅ } ↔ 𝑥 = ∅ )
4	3	bicomi	⊢ ( 𝑥 = ∅ ↔ 𝑥 ∈ { ∅ } )
5	4	necon3abii	⊢ ( 𝑥 ≠ ∅ ↔ ¬ 𝑥 ∈ { ∅ } )
6	2 5	anbi12i	⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) ↔ ( 𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ { ∅ } ) )
7		eldif	⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∖ { ∅ } ) ↔ ( 𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ { ∅ } ) )
8	6 7	bitr4i	⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) ↔ 𝑥 ∈ ( 𝒫 𝐴 ∖ { ∅ } ) )
9	8	imbi1i	⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) ↔ ( 𝑥 ∈ ( 𝒫 𝐴 ∖ { ∅ } ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) )
10	9	albii	⊢ ( ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) ↔ ∀ 𝑥 ( 𝑥 ∈ ( 𝒫 𝐴 ∖ { ∅ } ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) )
11		df-fr	⊢ ( 𝑅 Fr 𝐴 ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) )
12		df-ral	⊢ ( ∀ 𝑥 ∈ ( 𝒫 𝐴 ∖ { ∅ } ) ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ↔ ∀ 𝑥 ( 𝑥 ∈ ( 𝒫 𝐴 ∖ { ∅ } ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) )
13	10 11 12	3bitr4i	⊢ ( 𝑅 Fr 𝐴 ↔ ∀ 𝑥 ∈ ( 𝒫 𝐴 ∖ { ∅ } ) ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 )