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Mirrors > Home > MPE Home > Th. List > wereu | Structured version Visualization version GIF version |
Description: A subset of a well-ordered set has a unique 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 5509 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | fri 5481 | . . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | |
3 | 2 | exp32 424 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑅 Fr 𝐴) → (𝐵 ⊆ 𝐴 → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))) |
4 | 3 | expcom 417 | . . . 4 ⊢ (𝑅 Fr 𝐴 → (𝐵 ∈ 𝑉 → (𝐵 ⊆ 𝐴 → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)))) |
5 | 4 | 3imp2 1346 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
6 | 1, 5 | sylan 583 | . 2 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
7 | weso 5510 | . . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
8 | soss 5457 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Or 𝐴 → 𝑅 Or 𝐵)) | |
9 | 7, 8 | mpan9 510 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝑅 Or 𝐵) |
10 | somo 5474 | . . . 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 1186 | . 2 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) |
13 | reu5 3375 | . 2 ⊢ (∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) | |
14 | 6, 12, 13 | sylanbrc 586 | 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 399 ∧ w3a 1084 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∃!wreu 3108 ∃*wrmo 3109 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 Or wor 5437 Fr wfr 5475 We wwe 5477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-ex 1782 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 |
This theorem is referenced by: htalem 9309 zorn2lem1 9907 dyadmax 24202 finorwe 34799 wessf1ornlem 41811 |
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