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Mirrors > Home > ILE Home > Th. List > wepo | GIF version |
Description: A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.) |
Ref | Expression |
---|---|
wepo | ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 Po 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wefr 4330 | . . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | frirrg 4322 | . . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) | |
3 | 1, 2 | syl3an1 1260 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) |
4 | 3 | 3expa 1192 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) |
5 | df-3an 969 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴)) | |
6 | df-wetr 4306 | . . . . . . . . . 10 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | |
7 | 6 | simprbi 273 | . . . . . . . . 9 ⊢ (𝑅 We 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
8 | 7 | adantr 274 | . . . . . . . 8 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
9 | 8 | r19.21bi 2552 | . . . . . . 7 ⊢ (((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
10 | 9 | r19.21bi 2552 | . . . . . 6 ⊢ ((((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
11 | 10 | anasss 397 | . . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
12 | 11 | r19.21bi 2552 | . . . 4 ⊢ ((((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
13 | 12 | anasss 397 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
14 | 5, 13 | sylan2b 285 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
15 | 4, 14 | ispod 4276 | 1 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 Po 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∧ w3a 967 ∈ wcel 2135 ∀wral 2442 class class class wbr 3976 Po wpo 4266 Fr wfr 4300 We wwe 4302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-po 4268 df-frfor 4303 df-frind 4304 df-wetr 4306 |
This theorem is referenced by: (None) |
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