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Theorem wetrep 5251
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))

Proof of Theorem wetrep
StepHypRef Expression
1 weso 5249 . . 3 ( E We 𝐴 → E Or 𝐴)
2 sotr 5201 . . 3 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
31, 2sylan 489 . 2 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
4 epel 5174 . . 3 (𝑥 E 𝑦𝑥𝑦)
5 epel 5174 . . 3 (𝑦 E 𝑧𝑦𝑧)
64, 5anbi12i 735 . 2 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
7 epel 5174 . 2 (𝑥 E 𝑧𝑥𝑧)
83, 6, 73imtr3g 284 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wcel 2131   class class class wbr 4796   E cep 5170   Or wor 5178   We wwe 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-eprel 5171  df-po 5179  df-so 5180  df-we 5219
This theorem is referenced by:  wefrc  5252  ordelord  5898
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