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Theorem dfom5 9101
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 9099 . . 3 (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥𝑦𝑥))
2 vex 3472 . . . 4 𝑦 ∈ V
32elintab 4862 . . 3 (𝑦 {𝑥 ∣ Lim 𝑥} ↔ ∀𝑥(Lim 𝑥𝑦𝑥))
41, 3bitr4i 281 . 2 (𝑦 ∈ ω ↔ 𝑦 {𝑥 ∣ Lim 𝑥})
54eqriv 2819 1 ω = {𝑥 ∣ Lim 𝑥}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536   = wceq 1538  wcel 2114  {cab 2800   cint 4851  Lim wlim 6170  ωcom 7565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-om 7566
This theorem is referenced by: (None)
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