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Theorem dfom5 9645
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). Theorem 1.23 of [Schloeder] p. 4. (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 9643 . . 3 (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥𝑦𝑥))
2 vex 3479 . . . 4 𝑦 ∈ V
32elintab 4963 . . 3 (𝑦 {𝑥 ∣ Lim 𝑥} ↔ ∀𝑥(Lim 𝑥𝑦𝑥))
41, 3bitr4i 278 . 2 (𝑦 ∈ ω ↔ 𝑦 {𝑥 ∣ Lim 𝑥})
54eqriv 2730 1 ω = {𝑥 ∣ Lim 𝑥}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540   = wceq 1542  wcel 2107  {cab 2710   cint 4951  Lim wlim 6366  ωcom 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7856
This theorem is referenced by: (None)
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