Theorem limom 4527

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 4520	. 2 ⊢ Ord ω
2		peano1 4508	. 2 ⊢ ∅ ∈ ω
3		vex 2689	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3	sucex 4415	. . . . . . . 8 ⊢ suc 𝑥 ∈ V
5	4	isseti 2694	. . . . . . 7 ⊢ ∃𝑧 𝑧 = suc 𝑥
6		peano2 4509	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
7	3	sucid 4339	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
8	6, 7	jctil 310	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))
9		eleq2 2203	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ suc 𝑥))
10		eleq1 2202	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω))
11	9, 10	anbi12d 464	. . . . . . . 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 1581	. . . . . 6 ⊢ ∃𝑧(𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
14	13	19.37aiv 1653	. . . . 5 ⊢ (𝑥 ∈ ω → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
15		eluni 3739	. . . . 5 ⊢ (𝑥 ∈ ∪ ω ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
16	14, 15	sylibr 133	. . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ∪ ω)
17	16	ssriv 3101	. . 3 ⊢ ω ⊆ ∪ ω
18		orduniss 4347	. . . 4 ⊢ (Ord ω → ∪ ω ⊆ ω)
19	1, 18	ax-mp 5	. . 3 ⊢ ∪ ω ⊆ ω
20	17, 19	eqssi 3113	. 2 ⊢ ω = ∪ ω
21		dflim2 4292	. 2 ⊢ (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ∪ ω))
22	1, 2, 20, 21	mpbir3an 1163	1 ⊢ Lim ω

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ⊆ wss 3071  ∅c0 3363  ∪ cuni 3736  Ord word 4284  Lim wlim 4286  suc csuc 4287  ωcom 4504

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-ilim 4291  df-suc 4293  df-iom 4505

This theorem is referenced by:  freccllem  6299  frecfcllem  6301  frecsuclem  6303