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Theorem wepo 4377
Description: A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
Assertion
Ref Expression
wepo ((𝑅 We 𝐴𝐴𝑉) → 𝑅 Po 𝐴)

Proof of Theorem wepo
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wefr 4376 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 frirrg 4368 . . . 4 ((𝑅 Fr 𝐴𝐴𝑉𝑥𝐴) → ¬ 𝑥𝑅𝑥)
31, 2syl3an1 1282 . . 3 ((𝑅 We 𝐴𝐴𝑉𝑥𝐴) → ¬ 𝑥𝑅𝑥)
433expa 1205 . 2 (((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 df-3an 982 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴))
6 df-wetr 4352 . . . . . . . . . 10 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
76simprbi 275 . . . . . . . . 9 (𝑅 We 𝐴 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
87adantr 276 . . . . . . . 8 ((𝑅 We 𝐴𝐴𝑉) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
98r19.21bi 2578 . . . . . . 7 (((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109r19.21bi 2578 . . . . . 6 ((((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1110anasss 399 . . . . 5 (((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1211r19.21bi 2578 . . . 4 ((((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1312anasss 399 . . 3 (((𝑅 We 𝐴𝐴𝑉) ∧ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
145, 13sylan2b 287 . 2 (((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 4322 1 ((𝑅 We 𝐴𝐴𝑉) → 𝑅 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  w3a 980  wcel 2160  wral 2468   class class class wbr 4018   Po wpo 4312   Fr wfr 4346   We wwe 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314  df-frfor 4349  df-frind 4350  df-wetr 4352
This theorem is referenced by: (None)
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