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Theorem dfom5 8430
 Description: ω is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
dfom5 ω = {𝑥 ∣ Lim 𝑥}

Proof of Theorem dfom5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elom3 8428 . . 3 (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥𝑦𝑥))
2 vex 3176 . . . 4 𝑦 ∈ V
32elintab 4422 . . 3 (𝑦 {𝑥 ∣ Lim 𝑥} ↔ ∀𝑥(Lim 𝑥𝑦𝑥))
41, 3bitr4i 266 . 2 (𝑦 ∈ ω ↔ 𝑦 {𝑥 ∣ Lim 𝑥})
54eqriv 2607 1 ω = {𝑥 ∣ Lim 𝑥}
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∩ cint 4410  Lim wlim 5641  ωcom 6957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958 This theorem is referenced by: (None)
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