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Theorem wepo 4219
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 4218 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 frirrg 4210 . . . 4 ((𝑅 Fr 𝐴𝐴𝑉𝑥𝐴) → ¬ 𝑥𝑅𝑥)
31, 2syl3an1 1217 . . 3 ((𝑅 We 𝐴𝐴𝑉𝑥𝐴) → ¬ 𝑥𝑅𝑥)
433expa 1149 . 2 (((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 df-3an 932 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴))
6 df-wetr 4194 . . . . . . . . . 10 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
76simprbi 271 . . . . . . . . 9 (𝑅 We 𝐴 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
87adantr 272 . . . . . . . 8 ((𝑅 We 𝐴𝐴𝑉) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
98r19.21bi 2479 . . . . . . 7 (((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109r19.21bi 2479 . . . . . 6 ((((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1110anasss 394 . . . . 5 (((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1211r19.21bi 2479 . . . 4 ((((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1312anasss 394 . . 3 (((𝑅 We 𝐴𝐴𝑉) ∧ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
145, 13sylan2b 283 . 2 (((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 4164 1 ((𝑅 We 𝐴𝐴𝑉) → 𝑅 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  w3a 930  wcel 1448  wral 2375   class class class wbr 3875   Po wpo 4154   Fr wfr 4188   We wwe 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-po 4156  df-frfor 4191  df-frind 4192  df-wetr 4194
This theorem is referenced by: (None)
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