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Theorem frpomin2 6341
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 9746 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 6340 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2 vex 3476 . . . . . 6 𝑥 ∈ V
32dfpred3 6310 . . . . 5 Pred(𝑅, 𝐵, 𝑥) = {𝑦𝐵𝑦𝑅𝑥}
43eqeq1i 2735 . . . 4 (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦𝐵𝑦𝑅𝑥} = ∅)
5 rabeq0 4383 . . . 4 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
64, 5bitri 274 . . 3 (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
76rexbii 3092 . 2 (∃𝑥𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
81, 7sylibr 233 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 394  w3a 1085   = wceq 1539  wne 2938  wral 3059  wrex 3068  {crab 3430  wss 3947  c0 4321   class class class wbr 5147   Po wpo 5585   Fr wfr 5627   Se wse 5628  Predcpred 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-po 5587  df-fr 5630  df-se 5631  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299
This theorem is referenced by:  frpoind  6342  tz6.26  6347  fpr1  8290
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