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Theorem frpomin2 6195
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 9365 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 6194 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2 vex 3412 . . . . . 6 𝑥 ∈ V
32dfpred3 6170 . . . . 5 Pred(𝑅, 𝐵, 𝑥) = {𝑦𝐵𝑦𝑅𝑥}
43eqeq1i 2742 . . . 4 (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦𝐵𝑦𝑅𝑥} = ∅)
5 rabeq0 4299 . . . 4 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
64, 5bitri 278 . . 3 (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
76rexbii 3170 . 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 1089   = wceq 1543  wne 2940  wral 3061  wrex 3062  {crab 3065  wss 3866  c0 4237   class class class wbr 5053   Po wpo 5466   Fr wfr 5506   Se wse 5507  Predcpred 6159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-po 5468  df-fr 5509  df-se 5510  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160
This theorem is referenced by:  frpoind  6196  fpr1  8043
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