Theorem dffr6 5636

Description: Alternate definition of df-fr 5633. See dffr5 35479 for a definition without dummy variables (but note that their equivalence uses ax-sep 5300). (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 4609	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
2	1	bicomi 223	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ 𝒫 𝐴)
3		velsn 4646	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	bicomi 223	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
5	4	necon3abii 2976	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 ∈ {∅})
6	2, 5	anbi12i 626	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
7		eldif 3954	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
8	6, 7	bitr4i 277	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) ↔ 𝑥 ∈ (𝒫 𝐴 ∖ {∅}))
9	8	imbi1i 348	. . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
10	9	albii 1813	. 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
11		df-fr 5633	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
12		df-ral 3051	. 2 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
13	10, 11, 12	3bitr4i 302	1 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059   ∖ cdif 3941   ⊆ wss 3944  ∅c0 4322  𝒫 cpw 4604  {csn 4630   class class class wbr 5149   Fr wfr 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-v 3463  df-dif 3947  df-ss 3961  df-pw 4606  df-sn 4631  df-fr 5633

This theorem is referenced by:  frd  5637