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Theorem wetrep 5678
Description: On a class well-ordered by membership, the membership predicate is transitive. (Contributed by NM, 22-Apr-1994.)
Assertion
Ref Expression
wetrep (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))

Proof of Theorem wetrep
StepHypRef Expression
1 weso 5676 . . 3 ( E We 𝐴 → E Or 𝐴)
2 sotr 5617 . . 3 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
31, 2sylan 580 . 2 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
4 epel 5587 . . 3 (𝑥 E 𝑦𝑥𝑦)
5 epel 5587 . . 3 (𝑦 E 𝑧𝑦𝑧)
64, 5anbi12i 628 . 2 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
7 epel 5587 . 2 (𝑥 E 𝑧𝑥𝑧)
83, 6, 73imtr3g 295 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wcel 2108   class class class wbr 5143   E cep 5583   Or wor 5591   We wwe 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-po 5592  df-so 5593  df-we 5639
This theorem is referenced by:  wefrc  5679  ordelord  6406
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