Theorem frpomin2 6332

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 9709 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 6331	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
2		vex 3461	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	dfpred3 6303	. . . . 5 ⊢ Pred(𝑅, 𝐵, 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥}
4	3	eqeq1i 2770	. . . 4 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅)
5		rabeq0 4345	. . . 4 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
6	4, 5	bitri 278	. . 3 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
7	6	rexbii 3112	. 2 ⊢ (∃𝑥 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
8	1, 7	sylibr 237	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 400   ∧ w3a 1101   = wceq 1563   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  {crab 3417   ⊆ wss 3907  ∅c0 4288   class class class wbr 5105   Po wpo 5558   Fr wfr 5602   Se wse 5603  Predcpred 6291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-po 5560  df-fr 5605  df-se 5606  df-xp 5658  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292

This theorem is referenced by:  frpoind  6333  tz6.26  6338  fpr1  8288