Theorem limom 4598

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 4591	. 2 ⊢ Ord ω
2		peano1 4578	. 2 ⊢ ∅ ∈ ω
3		vex 2733	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3	sucex 4483	. . . . . . . 8 ⊢ suc 𝑥 ∈ V
5	4	isseti 2738	. . . . . . 7 ⊢ ∃𝑧 𝑧 = suc 𝑥
6		peano2 4579	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
7	3	sucid 4402	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
8	6, 7	jctil 310	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))
9		eleq2 2234	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ suc 𝑥))
10		eleq1 2233	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω))
11	9, 10	anbi12d 470	. . . . . . . 8 ⊢ (𝑧 = suc 𝑥 → ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω) ↔ (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω)))
12	8, 11	syl5ibr 155	. . . . . . 7 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω)))
13	5, 12	eximii 1595	. . . . . 6 ⊢ ∃𝑧(𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
14	13	19.37aiv 1668	. . . . 5 ⊢ (𝑥 ∈ ω → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
15		eluni 3799	. . . . 5 ⊢ (𝑥 ∈ ∪ ω ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
16	14, 15	sylibr 133	. . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ∪ ω)
17	16	ssriv 3151	. . 3 ⊢ ω ⊆ ∪ ω
18		orduniss 4410	. . . 4 ⊢ (Ord ω → ∪ ω ⊆ ω)
19	1, 18	ax-mp 5	. . 3 ⊢ ∪ ω ⊆ ω
20	17, 19	eqssi 3163	. 2 ⊢ ω = ∪ ω
21		dflim2 4355	. 2 ⊢ (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ∪ ω))
22	1, 2, 20, 21	mpbir3an 1174	1 ⊢ Lim ω

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ⊆ wss 3121  ∅c0 3414  ∪ cuni 3796  Ord word 4347  Lim wlim 4349  suc csuc 4350  ωcom 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-iord 4351  df-ilim 4354  df-suc 4356  df-iom 4575

This theorem is referenced by:  freccllem  6381  frecfcllem  6383  frecsuclem  6385