Theorem limom 4647

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.

Step	Hyp	Ref	Expression
1		ordom 4640	. 2 ⊢ Ord ω
2		peano1 4627	. 2 ⊢ ∅ ∈ ω
3		vex 2763	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3	sucex 4532	. . . . . . . 8 ⊢ suc 𝑥 ∈ V
5	4	isseti 2768	. . . . . . 7 ⊢ ∃𝑧 𝑧 = suc 𝑥
6		peano2 4628	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
7	3	sucid 4449	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
8	6, 7	jctil 312	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))
9		eleq2 2257	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ suc 𝑥))
10		eleq1 2256	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω))
11	9, 10	anbi12d 473	. . . . . . . 8 ⊢ (𝑧 = suc 𝑥 → ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω) ↔ (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω)))
12	8, 11	imbitrrid 156	. . . . . . 7 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω)))
13	5, 12	eximii 1613	. . . . . 6 ⊢ ∃𝑧(𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
14	13	19.37aiv 1686	. . . . 5 ⊢ (𝑥 ∈ ω → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
15		eluni 3839	. . . . 5 ⊢ (𝑥 ∈ ∪ ω ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
16	14, 15	sylibr 134	. . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ∪ ω)
17	16	ssriv 3184	. . 3 ⊢ ω ⊆ ∪ ω
18		orduniss 4457	. . . 4 ⊢ (Ord ω → ∪ ω ⊆ ω)
19	1, 18	ax-mp 5	. . 3 ⊢ ∪ ω ⊆ ω
20	17, 19	eqssi 3196	. 2 ⊢ ω = ∪ ω
21		dflim2 4402	. 2 ⊢ (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ∪ ω))
22	1, 2, 20, 21	mpbir3an 1181	1 ⊢ Lim ω

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164   ⊆ wss 3154  ∅c0 3447  ∪ cuni 3836  Ord word 4394  Lim wlim 4396  suc csuc 4397  ωcom 4623

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-tr 4129  df-iord 4398  df-ilim 4401  df-suc 4403  df-iom 4624

This theorem is referenced by:  freccllem  6457  frecfcllem  6459  frecsuclem  6461