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Theorem limom 4382
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 4375 . 2 Ord ω
2 peano1 4363 . 2 ∅ ∈ ω
3 vex 2613 . . . . . . . . 9 𝑥 ∈ V
43sucex 4271 . . . . . . . 8 suc 𝑥 ∈ V
54isseti 2616 . . . . . . 7 𝑧 𝑧 = suc 𝑥
6 peano2 4364 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
73sucid 4200 . . . . . . . . 9 𝑥 ∈ suc 𝑥
86, 7jctil 305 . . . . . . . 8 (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))
9 eleq2 2146 . . . . . . . . 9 (𝑧 = suc 𝑥 → (𝑥𝑧𝑥 ∈ suc 𝑥))
10 eleq1 2145 . . . . . . . . 9 (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω))
119, 10anbi12d 457 . . . . . . . 8 (𝑧 = suc 𝑥 → ((𝑥𝑧𝑧 ∈ ω) ↔ (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω)))
128, 11syl5ibr 154 . . . . . . 7 (𝑧 = suc 𝑥 → (𝑥 ∈ ω → (𝑥𝑧𝑧 ∈ ω)))
135, 12eximii 1534 . . . . . 6 𝑧(𝑥 ∈ ω → (𝑥𝑧𝑧 ∈ ω))
141319.37aiv 1606 . . . . 5 (𝑥 ∈ ω → ∃𝑧(𝑥𝑧𝑧 ∈ ω))
15 eluni 3624 . . . . 5 (𝑥 ω ↔ ∃𝑧(𝑥𝑧𝑧 ∈ ω))
1614, 15sylibr 132 . . . 4 (𝑥 ∈ ω → 𝑥 ω)
1716ssriv 3012 . . 3 ω ⊆ ω
18 orduniss 4208 . . . 4 (Ord ω → ω ⊆ ω)
191, 18ax-mp 7 . . 3 ω ⊆ ω
2017, 19eqssi 3024 . 2 ω = ω
21 dflim2 4153 . 2 (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ω))
221, 2, 20, 21mpbir3an 1121 1 Lim ω
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wss 2982  c0 3267   cuni 3621  Ord word 4145  Lim wlim 4147  suc csuc 4148  ωcom 4359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-int 3657  df-tr 3896  df-iord 4149  df-ilim 4152  df-suc 4154  df-iom 4360
This theorem is referenced by:  freccllem  6072  frecfcllem  6074  frecsuclem  6076
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