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Theorem elom 7446
Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 8964. (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 2872 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21imbi2d 342 . . 3 (𝑦 = 𝐴 → ((Lim 𝑥𝑦𝑥) ↔ (Lim 𝑥𝐴𝑥)))
32albidv 1902 . 2 (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥𝑦𝑥) ↔ ∀𝑥(Lim 𝑥𝐴𝑥)))
4 df-om 7444 . 2 ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥𝑦𝑥)}
53, 4elrab2 3624 1 (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1523   = wceq 1525  wcel 2083  Oncon0 6073  Lim wlim 6074  ωcom 7443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-om 7444
This theorem is referenced by:  limomss  7448  ordom  7452  nnlim  7456  limom  7458  elom3  8964  dfom5b  32984
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