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Mirrors > Home > MPE Home > Th. List > Mathboxes > welb | Structured version Visualization version GIF version |
Description: A nonempty subset of a well-ordered set has a lower bound. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
welb | ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → (◡𝑅 Or 𝐵 ∧ ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wess 5663 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 We 𝐴 → 𝑅 We 𝐵)) | |
2 | 1 | impcom 408 | . . . . 5 ⊢ ((𝑅 We 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝑅 We 𝐵) |
3 | weso 5667 | . . . . 5 ⊢ (𝑅 We 𝐵 → 𝑅 Or 𝐵) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝑅 Or 𝐵) |
5 | cnvso 6287 | . . . 4 ⊢ (𝑅 Or 𝐵 ↔ ◡𝑅 Or 𝐵) | |
6 | 4, 5 | sylib 217 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐵 ⊆ 𝐴) → ◡𝑅 Or 𝐵) |
7 | 6 | 3ad2antr2 1189 | . 2 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ◡𝑅 Or 𝐵) |
8 | wefr 5666 | . . . . 5 ⊢ (𝑅 We 𝐵 → 𝑅 Fr 𝐵) | |
9 | 2, 8 | syl 17 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝑅 Fr 𝐵) |
10 | 9 | 3ad2antr2 1189 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → 𝑅 Fr 𝐵) |
11 | ssidd 4005 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ⊆ 𝐵) | |
12 | 11 | 3anim2i 1153 | . . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐵 ∧ 𝐵 ≠ ∅)) |
13 | 12 | adantl 482 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐵 ∧ 𝐵 ≠ ∅)) |
14 | frinfm 36598 | . . 3 ⊢ ((𝑅 Fr 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐵 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) | |
15 | 10, 13, 14 | syl2anc 584 | . 2 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
16 | 7, 15 | jca 512 | 1 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → (◡𝑅 Or 𝐵 ∧ ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 Or wor 5587 Fr wfr 5628 We wwe 5630 ◡ccnv 5675 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-cnv 5684 |
This theorem is referenced by: (None) |
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