Theorem epelon2 44662

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 7796. This is a weak form of epelg 5555 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 5555	. 2 ⊢ (𝐵 ∈ On → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	adantl 487	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146   class class class wbr 5105   E cep 5553  Oncon0 6368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-eprel 5554

This theorem is used by: (None)