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Theorem epelon2 43534
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 7795. This is a weak form of epelg 5585 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 5585 . 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 2108   class class class wbr 5143   E cep 5583  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584
This theorem is referenced by: (None)
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