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Theorem wetrep 4252
Description: An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
Assertion
Ref Expression
wetrep (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem wetrep
StepHypRef Expression
1 df-3an 949 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴))
2 df-wetr 4226 . . . . . . . . 9 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
32simprbi 273 . . . . . . . 8 ( E We 𝐴 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
43r19.21bi 2497 . . . . . . 7 (( E We 𝐴𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
54r19.21bi 2497 . . . . . 6 ((( E We 𝐴𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
65anasss 396 . . . . 5 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
76r19.21bi 2497 . . . 4 ((( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
87anasss 396 . . 3 (( E We 𝐴 ∧ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
91, 8sylan2b 285 . 2 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
10 epel 4184 . . 3 (𝑥 E 𝑦𝑥𝑦)
11 epel 4184 . . 3 (𝑦 E 𝑧𝑦𝑧)
1210, 11anbi12i 455 . 2 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
13 epel 4184 . 2 (𝑥 E 𝑧𝑥𝑧)
149, 12, 133imtr3g 203 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947  wcel 1465  wral 2393   class class class wbr 3899   E cep 4179   Fr wfr 4220   We wwe 4222
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-eprel 4181  df-wetr 4226
This theorem is referenced by:  wessep  4462
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