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Theorem epelon2 42954
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 7781. 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 480 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098   class class class wbr 5150   E cep 5583  Oncon0 6372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-eprel 5584
This theorem is referenced by: (None)
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