Theorem limom 2387

Description: Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Our proof, however, does not require the axiom of Infinity.

Assertion

Ref	Expression
limom	⊢ Lim ω

Proof of Theorem limom

Step	Hyp	Ref	Expression
1		ordom 2382	. 2 ⊢ Ord ω
2		ordeleqon 2241	. . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3		ordeirr 2217	. . . . . 6 ⊢ (Ord ω → ¬ ω ∈ ω)
4	1, 3	ax-mp 6	. . . . 5 ⊢ ¬ ω ∈ ω
5		elomg 2376	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ (Ord ω ∧ ∀x(Lim x → ω ∈ x))))
6		ordtri1 2231	. . . . . . . . . . . . . . 15 ⊢ ((Ord x ∧ Ord ω) → (x ⊆ ω ↔ ¬ ω ∈ x))
7	6	adantr 306	. . . . . . . . . . . . . 14 ⊢ (((Ord x ∧ Ord ω) ∧ ¬ Lim ω) → (x ⊆ ω ↔ ¬ ω ∈ x))
8		ordsseleq 2227	. . . . . . . . . . . . . . . 16 ⊢ ((Ord x ∧ Ord ω) → (x ⊆ ω ↔ (x ∈ ω ∨ x = ω)))
9	8	biimpd 135	. . . . . . . . . . . . . . 15 ⊢ ((Ord x ∧ Ord ω) → (x ⊆ ω → (x ∈ ω ∨ x = ω)))
10		nnlim 2385	. . . . . . . . . . . . . . . . 17 ⊢ (x ∈ ω → ¬ Lim x)
11	10	a1i 7	. . . . . . . . . . . . . . . 16 ⊢ (¬ Lim ω → (x ∈ ω → ¬ Lim x))
12		limeq 2211	. . . . . . . . . . . . . . . . . . 19 ⊢ (x = ω → (Lim x ↔ Lim ω))
13	12	biimpd 135	. . . . . . . . . . . . . . . . . 18 ⊢ (x = ω → (Lim x → Lim ω))
14	13	con3d 87	. . . . . . . . . . . . . . . . 17 ⊢ (x = ω → (¬ Lim ω → ¬ Lim x))
15	14	com12 13	. . . . . . . . . . . . . . . 16 ⊢ (¬ Lim ω → (x = ω → ¬ Lim x))
16	11, 15	jaod 329	. . . . . . . . . . . . . . 15 ⊢ (¬ Lim ω → ((x ∈ ω ∨ x = ω) → ¬ Lim x))
17	9, 16	sylan9 359	. . . . . . . . . . . . . 14 ⊢ (((Ord x ∧ Ord ω) ∧ ¬ Lim ω) → (x ⊆ ω → ¬ Lim x))
18	7, 17	sylbird 180	. . . . . . . . . . . . 13 ⊢ (((Ord x ∧ Ord ω) ∧ ¬ Lim ω) → (¬ ω ∈ x → ¬ Lim x))
19	18	a3d 70	. . . . . . . . . . . 12 ⊢ (((Ord x ∧ Ord ω) ∧ ¬ Lim ω) → (Lim x → ω ∈ x))
20	1, 19	mpan12 530	. . . . . . . . . . 11 ⊢ ((Ord x ∧ ¬ Lim ω) → (Lim x → ω ∈ x))
21		limord 2283	. . . . . . . . . . 11 ⊢ (Lim x → Ord x)
22	20, 21	sylan 343	. . . . . . . . . 10 ⊢ ((Lim x ∧ ¬ Lim ω) → (Lim x → ω ∈ x))
23	22	exp 291	. . . . . . . . 9 ⊢ (Lim x → (¬ Lim ω → (Lim x → ω ∈ x)))
24	23	pm2.43b 61	. . . . . . . 8 ⊢ (¬ Lim ω → (Lim x → ω ∈ x))
25	24	19.21aiv 943	. . . . . . 7 ⊢ (¬ Lim ω → ∀x(Lim x → ω ∈ x))
26	25, 1	jctil 240	. . . . . 6 ⊢ (¬ Lim ω → (Ord ω ∧ ∀x(Lim x → ω ∈ x)))
27	5, 26	syl5bir 184	. . . . 5 ⊢ (ω ∈ On → (¬ Lim ω → ω ∈ ω))
28	4, 27	mt3i 100	. . . 4 ⊢ (ω ∈ On → Lim ω)
29		limon 2342	. . . . 5 ⊢ Lim On
30		limeq 2211	. . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On))
31	29, 30	mpbiri 169	. . . 4 ⊢ (ω = On → Lim ω)
32	28, 31	jaoi 275	. . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω)
33	2, 32	sylbi 174	. 2 ⊢ (Ord ω → Lim ω)
34	1, 33	ax-mp 6	1 ⊢ Lim ω

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  Ord word 2198  Oncon0 2199  Lim wlim 2200  ωcom 2372

This theorem is referenced by:  peano2b 2388  peano1 2390  ssnlim 2407  inf5 3472  elom3 3478  omenps 3482  omensuc 3483  cardlim 3657

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373

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