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Theorem elom 7235
Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 8721. (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 2828 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21imbi2d 329 . . 3 (𝑦 = 𝐴 → ((Lim 𝑥𝑦𝑥) ↔ (Lim 𝑥𝐴𝑥)))
32albidv 1999 . 2 (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥𝑦𝑥) ↔ ∀𝑥(Lim 𝑥𝐴𝑥)))
4 df-om 7233 . 2 ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥𝑦𝑥)}
53, 4elrab2 3508 1 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2140  Oncon0 5885  Lim wlim 5886  ωcom 7232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rab 3060  df-v 3343  df-om 7233
This theorem is referenced by:  limomss  7237  ordom  7241  nnlim  7245  limom  7247  elom3  8721  dfom5b  32347
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