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Theorem frd 5593
Description: A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (deduction form). (Contributed by BJ, 16-Nov-2024.)
Hypotheses
Ref Expression
frd.fr (𝜑𝑅 Fr 𝐴)
frd.ss (𝜑𝐵𝐴)
frd.ex (𝜑𝐵𝑉)
frd.n0 (𝜑𝐵 ≠ ∅)
Assertion
Ref Expression
frd (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem frd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simpr 486 . . 3 ((𝜑𝑧 = 𝐵) → 𝑧 = 𝐵)
2 biidd 262 . . . 4 ((𝜑𝑧 = 𝐵) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑥))
31, 2raleqbidv 3320 . . 3 ((𝜑𝑧 = 𝐵) → (∀𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
41, 3rexeqbidv 3321 . 2 ((𝜑𝑧 = 𝐵) → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
5 frd.ex . . . 4 (𝜑𝐵𝑉)
6 frd.ss . . . 4 (𝜑𝐵𝐴)
75, 6elpwd 4567 . . 3 (𝜑𝐵 ∈ 𝒫 𝐴)
8 frd.n0 . . . 4 (𝜑𝐵 ≠ ∅)
9 nelsn 4627 . . . 4 (𝐵 ≠ ∅ → ¬ 𝐵 ∈ {∅})
108, 9syl 17 . . 3 (𝜑 → ¬ 𝐵 ∈ {∅})
117, 10eldifd 3922 . 2 (𝜑𝐵 ∈ (𝒫 𝐴 ∖ {∅}))
12 frd.fr . . 3 (𝜑𝑅 Fr 𝐴)
13 dffr6 5592 . . 3 (𝑅 Fr 𝐴 ↔ ∀𝑧 ∈ (𝒫 𝐴 ∖ {∅})∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥)
1412, 13sylib 217 . 2 (𝜑 → ∀𝑧 ∈ (𝒫 𝐴 ∖ {∅})∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥)
154, 11, 14rspcdv2 3577 1 (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  cdif 3908  wss 3911  c0 4283  𝒫 cpw 4561  {csn 4587   class class class wbr 5106   Fr wfr 5586
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 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-v 3448  df-dif 3914  df-in 3918  df-ss 3928  df-pw 4563  df-sn 4588  df-fr 5589
This theorem is referenced by:  fri  5594  frxp3  8084
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