Theorem frpomin2 6361

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 9790 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 6360	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
2		vex 3483	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	dfpred3 6331	. . . . 5 ⊢ Pred(𝑅, 𝐵, 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥}
4	3	eqeq1i 2741	. . . 4 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅)
5		rabeq0 4387	. . . 4 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
6	4, 5	bitri 275	. . 3 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
7	6	rexbii 3093	. 2 ⊢ (∃𝑥 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
8	1, 7	sylibr 234	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 395   ∧ w3a 1086   = wceq 1539   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3435   ⊆ wss 3950  ∅c0 4332   class class class wbr 5142   Po wpo 5589   Fr wfr 5633   Se wse 5634  Predcpred 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-po 5591  df-fr 5636  df-se 5637  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320

This theorem is referenced by:  frpoind  6362  tz6.26  6367  fpr1  8329