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Theorem limom 4465
Description: Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.)
Assertion
Ref Expression
limom Lim ω

Proof of Theorem limom
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordom 4458 . 2 Ord ω
2 peano1 4446 . 2 ∅ ∈ ω
3 vex 2644 . . . . . . . . 9 𝑥 ∈ V
43sucex 4353 . . . . . . . 8 suc 𝑥 ∈ V
54isseti 2649 . . . . . . 7 𝑧 𝑧 = suc 𝑥
6 peano2 4447 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
73sucid 4277 . . . . . . . . 9 𝑥 ∈ suc 𝑥
86, 7jctil 308 . . . . . . . 8 (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))
9 eleq2 2163 . . . . . . . . 9 (𝑧 = suc 𝑥 → (𝑥𝑧𝑥 ∈ suc 𝑥))
10 eleq1 2162 . . . . . . . . 9 (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω))
119, 10anbi12d 460 . . . . . . . 8 (𝑧 = suc 𝑥 → ((𝑥𝑧𝑧 ∈ ω) ↔ (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω)))
128, 11syl5ibr 155 . . . . . . 7 (𝑧 = suc 𝑥 → (𝑥 ∈ ω → (𝑥𝑧𝑧 ∈ ω)))
135, 12eximii 1549 . . . . . 6 𝑧(𝑥 ∈ ω → (𝑥𝑧𝑧 ∈ ω))
141319.37aiv 1621 . . . . 5 (𝑥 ∈ ω → ∃𝑧(𝑥𝑧𝑧 ∈ ω))
15 eluni 3686 . . . . 5 (𝑥 ω ↔ ∃𝑧(𝑥𝑧𝑧 ∈ ω))
1614, 15sylibr 133 . . . 4 (𝑥 ∈ ω → 𝑥 ω)
1716ssriv 3051 . . 3 ω ⊆ ω
18 orduniss 4285 . . . 4 (Ord ω → ω ⊆ ω)
191, 18ax-mp 7 . . 3 ω ⊆ ω
2017, 19eqssi 3063 . 2 ω = ω
21 dflim2 4230 . 2 (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ω))
221, 2, 20, 21mpbir3an 1131 1 Lim ω
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1299  wex 1436  wcel 1448  wss 3021  c0 3310   cuni 3683  Ord word 4222  Lim wlim 4224  suc csuc 4225  ωcom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-tr 3967  df-iord 4226  df-ilim 4229  df-suc 4231  df-iom 4443
This theorem is referenced by:  freccllem  6229  frecfcllem  6231  frecsuclem  6233
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