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Theorem wecmpep 5572
Description: The elements of a class well-ordered by membership 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 5571 . 2 ( E We 𝐴 → E Or 𝐴)
2 solin 5519 . . 3 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
3 epel 5489 . . . 4 (𝑥 E 𝑦𝑥𝑦)
4 biid 260 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 epel 5489 . . . 4 (𝑦 E 𝑥𝑦𝑥)
63, 4, 53orbi123i 1154 . . 3 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
72, 6sylib 217 . 2 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
81, 7sylan 579 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1084  wcel 2108   class class class wbr 5070   E cep 5485   Or wor 5493   We wwe 5534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-eprel 5486  df-so 5495  df-we 5537
This theorem is referenced by:  tz7.7  6277
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