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Theorem wecmpep 5030
 Description: The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)
Assertion
Ref Expression
wecmpep (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))

Proof of Theorem wecmpep
StepHypRef Expression
1 weso 5029 . 2 ( E We 𝐴 → E Or 𝐴)
2 solin 4982 . . 3 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
3 epel 4952 . . . 4 (𝑥 E 𝑦𝑥𝑦)
4 biid 250 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 epel 4952 . . . 4 (𝑦 E 𝑥𝑦𝑥)
63, 4, 53orbi123i 1245 . . 3 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
72, 6sylib 207 . 2 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
81, 7sylan 487 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∨ w3o 1030   ∈ wcel 1977   class class class wbr 4583   E cep 4947   Or wor 4958   We wwe 4996 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-so 4960  df-we 4999 This theorem is referenced by:  tz7.7  5666
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