Theorem frpomin2 6343

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 9744 enables avoiding the axiom of 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 6342	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
2		vex 3479	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	dfpred3 6312	. . . . 5 ⊢ Pred(𝑅, 𝐵, 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥}
4	3	eqeq1i 2738	. . . 4 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅)
5		rabeq0 4385	. . . 4 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
6	4, 5	bitri 275	. . 3 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
7	6	rexbii 3095	. 2 ⊢ (∃𝑥 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
8	1, 7	sylibr 233	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 397   ∧ w3a 1088   = wceq 1542   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3433   ⊆ wss 3949  ∅c0 4323   class class class wbr 5149   Po wpo 5587   Fr wfr 5629   Se wse 5630  Predcpred 6300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-fr 5632  df-se 5633  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301

This theorem is referenced by:  frpoind  6344  tz6.26  6349  fpr1  8288