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Mirrors > Home > MPE Home > Th. List > frpomin2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
frpomin2 | ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frpomin 6340 | . 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | |
2 | vex 3476 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | dfpred3 6310 | . . . . 5 ⊢ Pred(𝑅, 𝐵, 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} |
4 | 3 | eqeq1i 2735 | . . . 4 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅) |
5 | rabeq0 4383 | . . . 4 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | |
6 | 4, 5 | bitri 274 | . . 3 ⊢ (Pred(𝑅, 𝐵, 𝑥) = ∅ ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
7 | 6 | rexbii 3092 | . 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) = ∅) |
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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