Theorem elom 7890

Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9662. (Contributed by NM, 15-May-1994.)

Assertion

Ref	Expression
elom	⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem elom

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2852	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2	1	imbi2d 343	. . 3 ⊢ (𝑦 = 𝐴 → ((Lim 𝑥 → 𝑦 ∈ 𝑥) ↔ (Lim 𝑥 → 𝐴 ∈ 𝑥)))
3	2	albidv 1954	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))
4		df-om 7888	. 2 ⊢ ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥)}
5	3, 4	elrab2 3652	1 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2146  Oncon0 6369  Lim wlim 6370  ωcom 7887

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

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-om 7888

This theorem is used by:  limomss  7892  trom  7896  nnlim  7901  limom  7903  peano1  7910  1onn  8663  2onn  8665  elom3  9662  dfom5b  36804