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Theorem wetrep 4375
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 982 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) ↔ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴))
2 df-wetr 4349 . . . . . . . . 9 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
32simprbi 275 . . . . . . . 8 ( E We 𝐴 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
43r19.21bi 2578 . . . . . . 7 (( E We 𝐴𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
54r19.21bi 2578 . . . . . 6 ((( E We 𝐴𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
65anasss 399 . . . . 5 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
76r19.21bi 2578 . . . 4 ((( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
87anasss 399 . . 3 (( E We 𝐴 ∧ ((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
91, 8sylan2b 287 . 2 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
10 epel 4307 . . 3 (𝑥 E 𝑦𝑥𝑦)
11 epel 4307 . . 3 (𝑦 E 𝑧𝑦𝑧)
1210, 11anbi12i 460 . 2 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
13 epel 4307 . 2 (𝑥 E 𝑧𝑥𝑧)
149, 12, 133imtr3g 204 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980  wcel 2160  wral 2468   class class class wbr 4018   E cep 4302   Fr wfr 4343   We wwe 4345
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-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-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-eprel 4304  df-wetr 4349
This theorem is referenced by:  wessep  4592
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