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Theorem elom 7268
Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 8762. (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 2832 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21imbi2d 331 . . 3 (𝑦 = 𝐴 → ((Lim 𝑥𝑦𝑥) ↔ (Lim 𝑥𝐴𝑥)))
32albidv 2015 . 2 (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥𝑦𝑥) ↔ ∀𝑥(Lim 𝑥𝐴𝑥)))
4 df-om 7266 . 2 ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥𝑦𝑥)}
53, 4elrab2 3525 1 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  Oncon0 5910  Lim wlim 5911  ωcom 7265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-om 7266
This theorem is referenced by:  limomss  7270  ordom  7274  nnlim  7278  limom  7280  elom3  8762  dfom5b  32466
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