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Theorem frpomin2 33262
 Description: Every (possibly proper) subclass of a class 𝐴 with a founded, partial-ordering, set-like relation 𝑅 has a minimal element. The additional condition of partial ordering over frmin 33267 enables avoiding infinity. (Contributed by Scott Fenton, 11-Feb-2022.)
Assertion
Ref Expression
frpomin2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅   𝑥,𝐵

Proof of Theorem frpomin2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 frpomin 33261 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2 vex 3445 . . . . . 6 𝑥 ∈ V
32dfpred3 6133 . . . . 5 Pred(𝑅, 𝐵, 𝑥) = {𝑦𝐵𝑦𝑅𝑥}
43eqeq1i 2803 . . . 4 (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦𝐵𝑦𝑅𝑥} = ∅)
5 rabeq0 4295 . . . 4 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
64, 5bitri 278 . . 3 (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
76rexbii 3211 . 2 (∃𝑥𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
81, 7sylibr 237 1 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3883  ∅c0 4246   class class class wbr 5034   Po wpo 5440   Fr wfr 5479   Se wse 5480  Predcpred 6122 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-po 5442  df-fr 5482  df-se 5483  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123 This theorem is referenced by:  frpoind  33263  fpr1  33322
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