![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fri | Structured version Visualization version GIF version |
Description: A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (inference form). (Contributed by BJ, 16-Nov-2024.) (Proof shortened by BJ, 19-Nov-2024.) |
Ref | Expression |
---|---|
fri | ⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 769 | . 2 ⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → 𝑅 Fr 𝐴) | |
2 | simprl 771 | . 2 ⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → 𝐵 ⊆ 𝐴) | |
3 | simpll 767 | . 2 ⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → 𝐵 ∈ 𝐶) | |
4 | simprr 773 | . 2 ⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → 𝐵 ≠ ∅) | |
5 | 1, 2, 3, 4 | frd 5644 | 1 ⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 Fr wfr 5637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-v 3479 df-dif 3965 df-ss 3979 df-pw 4606 df-sn 4631 df-fr 5640 |
This theorem is referenced by: frc 5651 fr2nr 5665 frminex 5667 wereu 5684 wereu2 5685 frpomin 6362 fr3nr 7790 frfi 9318 fimax2g 9319 fimin2g 9534 wofib 9582 wemapso 9588 wemapso2lem 9589 noinfep 9697 cflim2 10300 isfin1-3 10423 fin12 10450 fpwwe2lem11 10678 fpwwe2lem12 10679 fpwwe2 10680 bnj110 34850 frinfm 37721 fdc 37731 fnwe2lem2 43039 |
Copyright terms: Public domain | W3C validator |