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Theorem epelon2 43510
Description: Over the ordinal numbers, one may define the relation 𝐴 E 𝐵 iff 𝐴𝐵 and one finds that, under this ordering, On is a well-ordered class, see epweon 7793. This is a weak form of epelg 5589 which only requires that we know 𝐵 to be a set. (Contributed by RP, 27-Sep-2023.)
Assertion
Ref Expression
epelon2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵𝐴𝐵))

Proof of Theorem epelon2
StepHypRef Expression
1 epelg 5589 . 2 (𝐵 ∈ On → (𝐴 E 𝐵𝐴𝐵))
21adantl 481 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2105   class class class wbr 5147   E cep 5587  Oncon0 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-eprel 5588
This theorem is referenced by: (None)
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