Theorem frpomin2 32600

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 32605 enables avoiding infinity. (Contributed by Scott Fenton, 11-Feb-2022.)

Assertion

Ref	Expression
frpomin2	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 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.

Step	Hyp	Ref	Expression
1		frpomin 32599	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
2		vex 3418	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	dfpred3 5998	. . . . 5 ⊢ Pred(𝑅, 𝐵, 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥}
4	3	eqeq1i 2783	. . . 4 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅)
5		rabeq0 4226	. . . 4 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
6	4, 5	bitri 267	. . 3 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
7	6	rexbii 3194	. 2 ⊢ (∃𝑥 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
8	1, 7	sylibr 226	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ≠ wne 2967  ∀wral 3088  ∃wrex 3089  {crab 3092   ⊆ wss 3831  ∅c0 4180   class class class wbr 4930   Po wpo 5325   Fr wfr 5364   Se wse 5365  Predcpred 5987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-po 5327  df-fr 5367  df-se 5368  df-xp 5414  df-cnv 5416  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988

This theorem is referenced by:  frpoind  32601  fpr1  32660