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Theorem wetrep 5541
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 5539 . . 3 ( E We 𝐴 → E Or 𝐴)
2 sotr 5490 . . 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 5462 . . 3 (𝑥 E 𝑦𝑥𝑦)
5 epel 5462 . . 3 (𝑦 E 𝑧𝑦𝑧)
64, 5anbi12i 626 . 2 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
7 epel 5462 . 2 (𝑥 E 𝑧𝑥𝑧)
83, 6, 73imtr3g 296 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wcel 2105   class class class wbr 5057   E cep 5457   Or wor 5466   We wwe 5506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-eprel 5458  df-po 5467  df-so 5468  df-we 5509
This theorem is referenced by:  wefrc  5542  ordelord  6206
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