Theorem epelon2 43965

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 7718. This is a weak form of epelg 5519 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 5519	. 2 ⊢ (𝐵 ∈ On → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	adantl 482	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119   class class class wbr 5072   E cep 5517  Oncon0 6310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-eprel 5518

This theorem is referenced by: (None)