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Theorem dffr6 5635
Description: Alternate definition of df-fr 5632. See dffr5 34724 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 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Proof of Theorem dffr6
StepHypRef Expression
1 velpw 4608 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
21bicomi 223 . . . . . 6 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
3 velsn 4645 . . . . . . . 8 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43bicomi 223 . . . . . . 7 (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
54necon3abii 2988 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 ∈ {∅})
62, 5anbi12i 628 . . . . 5 ((𝑥𝐴𝑥 ≠ ∅) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
7 eldif 3959 . . . . 5 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
86, 7bitr4i 278 . . . 4 ((𝑥𝐴𝑥 ≠ ∅) ↔ 𝑥 ∈ (𝒫 𝐴 ∖ {∅}))
98imbi1i 350 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
109albii 1822 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
11 df-fr 5632 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
12 df-ral 3063 . 2 (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
1310, 11, 123bitr4i 303 1 (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  cdif 3946  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629   class class class wbr 5149   Fr wfr 5629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-pw 4605  df-sn 4630  df-fr 5632
This theorem is referenced by:  frd  5636
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