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Theorem wepo 4122
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 4121 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 frirrg 4113 . . . 4 ((𝑅 Fr 𝐴𝐴𝑉𝑥𝐴) → ¬ 𝑥𝑅𝑥)
31, 2syl3an1 1203 . . 3 ((𝑅 We 𝐴𝐴𝑉𝑥𝐴) → ¬ 𝑥𝑅𝑥)
433expa 1139 . 2 (((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 df-3an 922 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴))
6 df-wetr 4097 . . . . . . . . . 10 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
76simprbi 269 . . . . . . . . 9 (𝑅 We 𝐴 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
87adantr 270 . . . . . . . 8 ((𝑅 We 𝐴𝐴𝑉) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
98r19.21bi 2450 . . . . . . 7 (((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109r19.21bi 2450 . . . . . 6 ((((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1110anasss 391 . . . . 5 (((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1211r19.21bi 2450 . . . 4 ((((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1312anasss 391 . . 3 (((𝑅 We 𝐴𝐴𝑉) ∧ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
145, 13sylan2b 281 . 2 (((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 4067 1 ((𝑅 We 𝐴𝐴𝑉) → 𝑅 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  w3a 920  wcel 1434  wral 2349   class class class wbr 3793   Po wpo 4057   Fr wfr 4091   We wwe 4093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-po 4059  df-frfor 4094  df-frind 4095  df-wetr 4097
This theorem is referenced by: (None)
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