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Theorem limom 4631
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 4624 . 2 Ord ω
2 peano1 4611 . 2 ∅ ∈ ω
3 vex 2755 . . . . . . . . 9 𝑥 ∈ V
43sucex 4516 . . . . . . . 8 suc 𝑥 ∈ V
54isseti 2760 . . . . . . 7 𝑧 𝑧 = suc 𝑥
6 peano2 4612 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
73sucid 4435 . . . . . . . . 9 𝑥 ∈ suc 𝑥
86, 7jctil 312 . . . . . . . 8 (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))
9 eleq2 2253 . . . . . . . . 9 (𝑧 = suc 𝑥 → (𝑥𝑧𝑥 ∈ suc 𝑥))
10 eleq1 2252 . . . . . . . . 9 (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω))
119, 10anbi12d 473 . . . . . . . 8 (𝑧 = suc 𝑥 → ((𝑥𝑧𝑧 ∈ ω) ↔ (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω)))
128, 11imbitrrid 156 . . . . . . 7 (𝑧 = suc 𝑥 → (𝑥 ∈ ω → (𝑥𝑧𝑧 ∈ ω)))
135, 12eximii 1613 . . . . . 6 𝑧(𝑥 ∈ ω → (𝑥𝑧𝑧 ∈ ω))
141319.37aiv 1686 . . . . 5 (𝑥 ∈ ω → ∃𝑧(𝑥𝑧𝑧 ∈ ω))
15 eluni 3827 . . . . 5 (𝑥 ω ↔ ∃𝑧(𝑥𝑧𝑧 ∈ ω))
1614, 15sylibr 134 . . . 4 (𝑥 ∈ ω → 𝑥 ω)
1716ssriv 3174 . . 3 ω ⊆ ω
18 orduniss 4443 . . . 4 (Ord ω → ω ⊆ ω)
191, 18ax-mp 5 . . 3 ω ⊆ ω
2017, 19eqssi 3186 . 2 ω = ω
21 dflim2 4388 . 2 (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ω))
221, 2, 20, 21mpbir3an 1181 1 Lim ω
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wex 1503  wcel 2160  wss 3144  c0 3437   cuni 3824  Ord word 4380  Lim wlim 4382  suc csuc 4383  ωcom 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-tr 4117  df-iord 4384  df-ilim 4387  df-suc 4389  df-iom 4608
This theorem is referenced by:  freccllem  6428  frecfcllem  6430  frecsuclem  6432
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