Theorem limom 4662

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 4655	. 2 ⊢ Ord ω
2		peano1 4642	. 2 ⊢ ∅ ∈ ω
3		vex 2775	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3	sucex 4547	. . . . . . . 8 ⊢ suc 𝑥 ∈ V
5	4	isseti 2780	. . . . . . 7 ⊢ ∃𝑧 𝑧 = suc 𝑥
6		peano2 4643	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
7	3	sucid 4464	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
8	6, 7	jctil 312	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))
9		eleq2 2269	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ suc 𝑥))
10		eleq1 2268	. . . . . . . . 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 1625	. . . . . 6 ⊢ ∃𝑧(𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
14	13	19.37aiv 1698	. . . . 5 ⊢ (𝑥 ∈ ω → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
15		eluni 3853	. . . . 5 ⊢ (𝑥 ∈ ∪ ω ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))
16	14, 15	sylibr 134	. . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ∪ ω)
17	16	ssriv 3197	. . 3 ⊢ ω ⊆ ∪ ω
18		orduniss 4472	. . . 4 ⊢ (Ord ω → ∪ ω ⊆ ω)
19	1, 18	ax-mp 5	. . 3 ⊢ ∪ ω ⊆ ω
20	17, 19	eqssi 3209	. 2 ⊢ ω = ∪ ω
21		dflim2 4417	. 2 ⊢ (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ∪ ω))
22	1, 2, 20, 21	mpbir3an 1182	1 ⊢ Lim ω

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1515   ∈ wcel 2176   ⊆ wss 3166  ∅c0 3460  ∪ cuni 3850  Ord word 4409  Lim wlim 4411  suc csuc 4412  ωcom 4638

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-tr 4143  df-iord 4413  df-ilim 4416  df-suc 4418  df-iom 4639

This theorem is referenced by:  freccllem  6488  frecfcllem  6490  frecsuclem  6492