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Theorem frpomin2 6364
Description: Every nonempty (possibly proper) subclass of a class 𝐴 with a well-founded set-like partial order 𝑅 has a minimal element. The additional condition of partial order over frmin 9787 enables avoiding the axiom of 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 6363 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2 vex 3482 . . . . . 6 𝑥 ∈ V
32dfpred3 6334 . . . . 5 Pred(𝑅, 𝐵, 𝑥) = {𝑦𝐵𝑦𝑅𝑥}
43eqeq1i 2740 . . . 4 (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦𝐵𝑦𝑅𝑥} = ∅)
5 rabeq0 4394 . . . 4 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
64, 5bitri 275 . . 3 (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
76rexbii 3092 . 2 (∃𝑥𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
81, 7sylibr 234 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 395  w3a 1086   = wceq 1537  wne 2938  wral 3059  wrex 3068  {crab 3433  wss 3963  c0 4339   class class class wbr 5148   Po wpo 5595   Fr wfr 5638   Se wse 5639  Predcpred 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-po 5597  df-fr 5641  df-se 5642  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323
This theorem is referenced by:  frpoind  6365  tz6.26  6370  fpr1  8327
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