Theorem dffr6 5619

Description: Alternate definition of df-fr 5616. See dffr5 36227 for a definition without dummy variables (but note that their equivalence uses ax-sep 5258). (Contributed by BJ, 16-Nov-2024.)

Assertion

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

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Proof of Theorem dffr6

Step	Hyp	Ref	Expression
1		velpw 4568	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
2	1	bicomi 227	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ 𝒫 𝐴)
3		velsn 4606	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	bicomi 227	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
5	4	necon3abii 3004	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 ∈ {∅})
6	2, 5	anbi12i 639	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
7		eldif 3916	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
8	6, 7	bitr4i 281	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ 𝑥 ∈ (𝒫 𝐴 ∖ {∅}))
9	8	imbi1i 352	. . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
10	9	albii 1849	. 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
11		df-fr 5616	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
12		df-ral 3080	. 2 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
13	10, 11, 12	3bitr4i 306	1 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {csn 4590   class class class wbr 5110   Fr wfr 5613

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-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-v 3457  df-dif 3909  df-ss 3923  df-pw 4565  df-sn 4591  df-fr 5616

This theorem is referenced by:  frd  5620