Theorem elom 7890

Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9688. (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 2829	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2	1	imbi2d 340	. . 3 ⊢ (𝑦 = 𝐴 → ((Lim 𝑥 → 𝑦 ∈ 𝑥) ↔ (Lim 𝑥 → 𝐴 ∈ 𝑥)))
3	2	albidv 1920	. 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 3695	1 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  Oncon0 6384  Lim wlim 6385  ωcom 7887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-om 7888

This theorem is referenced by:  limomss  7892  trom  7896  nnlim  7901  limom  7903  peano1  7910  1onn  8678  2onn  8680  elom3  9688  dfom5b  35913