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Theorem wecmpep 5302
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 5301 . 2 ( E We 𝐴 → E Or 𝐴)
2 solin 5254 . . 3 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
3 epel 5226 . . . 4 (𝑥 E 𝑦𝑥𝑦)
4 biid 253 . . . 4 (𝑥 = 𝑦𝑥 = 𝑦)
5 epel 5226 . . . 4 (𝑦 E 𝑥𝑦𝑥)
63, 4, 53orbi123i 1196 . . 3 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
72, 6sylib 210 . 2 (( E Or 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
81, 7sylan 576 1 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3o 1107  wcel 2157   class class class wbr 4841   E cep 5222   Or wor 5230   We wwe 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-eprel 5223  df-so 5232  df-we 5271
This theorem is referenced by:  tz7.7  5965
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