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| 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 5641 | 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 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 Fr wfr 5634 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-pw 4602 df-sn 4627 df-fr 5637 |
| This theorem is referenced by: frc 5648 fr2nr 5662 frminex 5664 wereu 5681 wereu2 5682 frpomin 6361 fr3nr 7792 frfi 9321 fimax2g 9322 fimin2g 9537 wofib 9585 wemapso 9591 wemapso2lem 9592 noinfep 9700 cflim2 10303 isfin1-3 10426 fin12 10453 fpwwe2lem11 10681 fpwwe2lem12 10682 fpwwe2 10683 bnj110 34872 frinfm 37742 fdc 37752 fnwe2lem2 43063 sswfaxreg 45004 |
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