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Theorem dfom4 9506
Description: A simplification of df-om 7781 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
Assertion
Ref Expression
dfom4 ω = {𝑥 ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfom4
StepHypRef Expression
1 elom3 9505 . 2 (𝑥 ∈ ω ↔ ∀𝑦(Lim 𝑦𝑥𝑦))
21abbi2i 2877 1 ω = {𝑥 ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  {cab 2713  Lim wlim 6303  ωcom 7780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650  ax-inf2 9498
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-tr 5210  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-om 7781
This theorem is referenced by: (None)
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