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Theorem elom 7570
 Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9102. (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 2877 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21imbi2d 344 . . 3 (𝑦 = 𝐴 → ((Lim 𝑥𝑦𝑥) ↔ (Lim 𝑥𝐴𝑥)))
32albidv 1921 . 2 (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥𝑦𝑥) ↔ ∀𝑥(Lim 𝑥𝐴𝑥)))
4 df-om 7568 . 2 ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥𝑦𝑥)}
53, 4elrab2 3631 1 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Oncon0 6162  Lim wlim 6163  ωcom 7567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-om 7568 This theorem is referenced by:  limomss  7572  ordom  7576  nnlim  7580  limom  7582  elom3  9102  dfom5b  33522
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