MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffr6 Structured version   Visualization version   GIF version

Theorem dffr6 5633
Description: Alternate definition of df-fr 5630. See dffr5 34712 for a definition without dummy variables (but note that their equivalence uses ax-sep 5298). (Contributed by BJ, 16-Nov-2024.)
Assertion
Ref Expression
dffr6 (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Proof of Theorem dffr6
StepHypRef Expression
1 velpw 4606 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
21bicomi 223 . . . . . 6 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
3 velsn 4643 . . . . . . . 8 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43bicomi 223 . . . . . . 7 (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
54necon3abii 2987 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 ∈ {∅})
62, 5anbi12i 627 . . . . 5 ((𝑥𝐴𝑥 ≠ ∅) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
7 eldif 3957 . . . . 5 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
86, 7bitr4i 277 . . . 4 ((𝑥𝐴𝑥 ≠ ∅) ↔ 𝑥 ∈ (𝒫 𝐴 ∖ {∅}))
98imbi1i 349 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
109albii 1821 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
11 df-fr 5630 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
12 df-ral 3062 . 2 (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
1310, 11, 123bitr4i 302 1 (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  cdif 3944  wss 3947  c0 4321  𝒫 cpw 4601  {csn 4627   class class class wbr 5147   Fr wfr 5627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-v 3476  df-dif 3950  df-in 3954  df-ss 3964  df-pw 4603  df-sn 4628  df-fr 5630
This theorem is referenced by:  frd  5634
  Copyright terms: Public domain W3C validator