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Theorem limom 4361
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 4354 . 2 Ord ω
2 peano1 4342 . 2 ∅ ∈ ω
3 vex 2575 . . . . . . . . 9 𝑥 ∈ V
43sucex 4250 . . . . . . . 8 suc 𝑥 ∈ V
54isseti 2578 . . . . . . 7 𝑧 𝑧 = suc 𝑥
6 peano2 4343 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
73sucid 4179 . . . . . . . . 9 𝑥 ∈ suc 𝑥
86, 7jctil 299 . . . . . . . 8 (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))
9 eleq2 2115 . . . . . . . . 9 (𝑧 = suc 𝑥 → (𝑥𝑧𝑥 ∈ suc 𝑥))
10 eleq1 2114 . . . . . . . . 9 (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω))
119, 10anbi12d 450 . . . . . . . 8 (𝑧 = suc 𝑥 → ((𝑥𝑧𝑧 ∈ ω) ↔ (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω)))
128, 11syl5ibr 149 . . . . . . 7 (𝑧 = suc 𝑥 → (𝑥 ∈ ω → (𝑥𝑧𝑧 ∈ ω)))
135, 12eximii 1507 . . . . . 6 𝑧(𝑥 ∈ ω → (𝑥𝑧𝑧 ∈ ω))
141319.37aiv 1579 . . . . 5 (𝑥 ∈ ω → ∃𝑧(𝑥𝑧𝑧 ∈ ω))
15 eluni 3608 . . . . 5 (𝑥 ω ↔ ∃𝑧(𝑥𝑧𝑧 ∈ ω))
1614, 15sylibr 141 . . . 4 (𝑥 ∈ ω → 𝑥 ω)
1716ssriv 2974 . . 3 ω ⊆ ω
18 orduniss 4187 . . . 4 (Ord ω → ω ⊆ ω)
191, 18ax-mp 7 . . 3 ω ⊆ ω
2017, 19eqssi 2986 . 2 ω = ω
21 dflim2 4132 . 2 (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ω))
221, 2, 20, 21mpbir3an 1095 1 Lim ω
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wex 1395  wcel 1407  wss 2942  c0 3249   cuni 3605  Ord word 4124  Lim wlim 4126  suc csuc 4127  ωcom 4338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-uni 3606  df-int 3641  df-tr 3880  df-iord 4128  df-ilim 4131  df-suc 4133  df-iom 4339
This theorem is referenced by: (None)
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