|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > wereu | Structured version Visualization version GIF version | ||
| Description: A nonempty subset of an 𝑅-well-ordered class has a unique 𝑅 -minimal element. (Contributed by NM, 18-Mar-1997.) (Revised by Mario Carneiro, 28-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| wereu | ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | wefr 5675 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
| 2 | fri 5642 | . . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | |
| 3 | 2 | exp32 420 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑅 Fr 𝐴) → (𝐵 ⊆ 𝐴 → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))) | 
| 4 | 3 | expcom 413 | . . . 4 ⊢ (𝑅 Fr 𝐴 → (𝐵 ∈ 𝑉 → (𝐵 ⊆ 𝐴 → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)))) | 
| 5 | 4 | 3imp2 1350 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | 
| 6 | 1, 5 | sylan 580 | . 2 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | 
| 7 | weso 5676 | . . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
| 8 | soss 5612 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Or 𝐴 → 𝑅 Or 𝐵)) | |
| 9 | 7, 8 | mpan9 506 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝑅 Or 𝐵) | 
| 10 | somo 5631 | . . . 4 ⊢ (𝑅 Or 𝐵 → ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | |
| 11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | 
| 12 | 11 | 3ad2antr2 1190 | . 2 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | 
| 13 | reu5 3382 | . 2 ⊢ (∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) | |
| 14 | 6, 12, 13 | sylanbrc 583 | 1 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∃!wreu 3378 ∃*wrmo 3379 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 Or wor 5591 Fr wfr 5634 We wwe 5636 | 
| 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-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 | 
| This theorem is referenced by: htalem 9936 zorn2lem1 10536 dyadmax 25633 wevgblacfn 35114 finorwe 37383 wessf1ornlem 45190 | 
| Copyright terms: Public domain | W3C validator |