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Theorem wecmpep 5523
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 5522 . 2 ( E We 𝐴 → E Or 𝐴)
2 solin 5474 . . 3 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
3 epel 5445 . . . 4 (𝑥 E 𝑦𝑥𝑦)
4 biid 263 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 epel 5445 . . . 4 (𝑦 E 𝑥𝑦𝑥)
63, 4, 53orbi123i 1152 . . 3 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
72, 6sylib 220 . 2 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
81, 7sylan 582 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3o 1082  wcel 2114   class class class wbr 5042   E cep 5440   Or wor 5449   We wwe 5489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-eprel 5441  df-so 5451  df-we 5492
This theorem is referenced by:  tz7.7  6193
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