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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 𝑥𝐴𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem elom
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2829 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21imbi2d 340 . . 3 (𝑦 = 𝐴 → ((Lim 𝑥𝑦𝑥) ↔ (Lim 𝑥𝐴𝑥)))
32albidv 1920 . 2 (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥𝑦𝑥) ↔ ∀𝑥(Lim 𝑥𝐴𝑥)))
4 df-om 7888 . 2 ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥𝑦𝑥)}
53, 4elrab2 3695 1 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
Colors of variables: wff setvar class
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
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