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Theorem wepo 4096
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 4095 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 frirrg 4087 . . . 4 ((𝑅 Fr 𝐴𝐴𝑉𝑥𝐴) → ¬ 𝑥𝑅𝑥)
31, 2syl3an1 1168 . . 3 ((𝑅 We 𝐴𝐴𝑉𝑥𝐴) → ¬ 𝑥𝑅𝑥)
433expa 1104 . 2 (((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) → ¬ 𝑥𝑅𝑥)
5 df-3an 887 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴))
6 df-wetr 4071 . . . . . . . . . 10 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
76simprbi 260 . . . . . . . . 9 (𝑅 We 𝐴 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
87adantr 261 . . . . . . . 8 ((𝑅 We 𝐴𝐴𝑉) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
98r19.21bi 2407 . . . . . . 7 (((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109r19.21bi 2407 . . . . . 6 ((((𝑅 We 𝐴𝐴𝑉) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1110anasss 379 . . . . 5 (((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1211r19.21bi 2407 . . . 4 ((((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1312anasss 379 . . 3 (((𝑅 We 𝐴𝐴𝑉) ∧ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
145, 13sylan2b 271 . 2 (((𝑅 We 𝐴𝐴𝑉) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
154, 14ispod 4041 1 ((𝑅 We 𝐴𝐴𝑉) → 𝑅 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  w3a 885  wcel 1393  wral 2306   class class class wbr 3764   Po wpo 4031   Fr wfr 4065   We wwe 4067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-frfor 4068  df-frind 4069  df-wetr 4071
This theorem is referenced by: (None)
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