Theorem epelon2 42829

Description: Over the ordinal numbers, one may define the relation 𝐴 E 𝐵 iff 𝐴 ∈ 𝐵 and one finds that, under this ordering, On is a well-ordered class, see epweon 7758. This is a weak form of epelg 5574 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 𝐵 ↔ 𝐴 ∈ 𝐵))

Proof of Theorem epelon2

Step	Hyp	Ref	Expression
1		epelg 5574	. 2 ⊢ (𝐵 ∈ On → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	adantl 481	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098   class class class wbr 5141   E cep 5572  Oncon0 6357

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-eprel 5573

This theorem is referenced by: (None)