MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wecmpep Structured version   Visualization version   GIF version

Theorem wecmpep 5681
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 5680 . 2 ( E We 𝐴 → E Or 𝐴)
2 solin 5623 . . 3 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
3 epel 5592 . . . 4 (𝑥 E 𝑦𝑥𝑦)
4 biid 261 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 epel 5592 . . . 4 (𝑦 E 𝑥𝑦𝑥)
63, 4, 53orbi123i 1155 . . 3 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
72, 6sylib 218 . 2 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
81, 7sylan 580 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1085  wcel 2106   class class class wbr 5148   E cep 5588   Or wor 5596   We wwe 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-so 5598  df-we 5643
This theorem is referenced by:  tz7.7  6412
  Copyright terms: Public domain W3C validator