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Theorem dfom5 8492
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 8490 . . 3 (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥𝑦𝑥))
2 vex 3194 . . . 4 𝑦 ∈ V
32elintab 4457 . . 3 (𝑦 {𝑥 ∣ Lim 𝑥} ↔ ∀𝑥(Lim 𝑥𝑦𝑥))
41, 3bitr4i 267 . 2 (𝑦 ∈ ω ↔ 𝑦 {𝑥 ∣ Lim 𝑥})
54eqriv 2623 1 ω = {𝑥 ∣ Lim 𝑥}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478   = wceq 1480  wcel 1992  {cab 2612   cint 4445  Lim wlim 5686  ωcom 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903  ax-inf2 8483
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-om 7014
This theorem is referenced by: (None)
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